This file was last updated: 11/17/2018 1:34:07 AM
========================================================== p. 41
Section C. More Algorithms>
Amplification of Section 3.
A scavenger hunt on OEIS for additional rules followed
in Table 5_14_18.
-------------------------------------------------
k = 1
There are no real solutions, so ignore.
-------------------------------------------------
k = 2. x = ( 2 + sqrt(4-4) ) / 2 = 1
2 2 2 2 2 2 2 2
-------------------------------------------------
k = 3. x = ( 3 + sqrt(5) ) / 2 = 2.618034
2 3 7 18 47 123 322 843 2207
A005248
A1: a(0)=2, a(1)=3, a(n) = 3*a(n-1) - a(n-2). -
Michael Somos, Jun 28 2003
A2: a(n) = ((3-sqrt(5))^n + (3+sqrt(5))^n)/2^n
A3: G.f.: (2-3*x)/(1-3*x+x^2). - Simon Plouffe in his 1992 dissertation.
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 7xy + y^2 + 45 = 0. - Colin Barker, Feb 16 2014
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 18xy + y^2 + 320 = 0. - Colin Barker, Feb 16 2014
A6: a(n) = S(n, 3) - S(n-2, 3) = 2*T(n, 3/2) with S(n-1, 3)
-------------------------------------------------
k = 4. x = ( 4 + sqrt(12) ) / 2 = 3.7320508
2, 4, 14, 52, 194, 724, 2702, 10084, 37634, 140452, 524174, 1956244,
A003500
A1: a(0) = 2, a(1) = 4; for n >= 2 a(n) = 4 * a(n-1) - a(n-2).
A2: a(n) = ( 2 + sqrt(3) )^n + ( 2 - sqrt(3) )^n.
A3: G.f.: -2*(-1+2*x)/(1-4*x+x^2).
A4: E.g.f.: 2*exp(2*x)*cosh(sqrt(3)*x). - Ilya Gutkovskiy, Apr 27 2016
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 4xy + y^2 + 12 = 0. - Colin Barker, Feb 04 2014
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 14xy + y^2 + 192 = 0. - Colin Barker, Feb 16 2014
A6:
AUTHOR N. J. A. Sloane. No date given.
-------------------------------------------------
k = 5. x = ( 5 + sqrt(21) ) / 2 = 4.7912818
2 5 23 110 527 2525 12098
A003501
A1: a(0) = 2, a(1) = 5; for n >= 2 a(n) = 5 * a(n-1) - a(n-2).
A2: a(n) = ap^n + am^n, with ap=(5+sqrt(21))/2 and am=(5-sqrt(21))/2.
A3: G.f.: (2-5*x)/(1-5*x+x^2). - Simon Plouffe in his 1992 dissertation.
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 5xy + y^2 + 21 = 0. - Colin Barker, Feb 08 2014
A6: a(n) = 5*S(n-1, 5) - 2*S(n-2, 5) = S(n, 5) - S(n-2, 5) = 2*T(n, 5/2)
AUTHOR N. J. A. Sloane. No date given.
-------------------------------------------------
k = 6. x = ( 6 + sqrt(32) ) / 2 = 5.8284277
2 6 34 198 1154 6726 39202
A003499
A1: a(0) = 2, a(1) = 6; for n >= 2 a(n) = 6 * a(n-1) - a(n-2).
A2: a(n) = (3+2*sqrt(2))^n+(3-2*sqrt(2))^n
A3: G.f.: (2-6*x)/(1-6*x+x^2).
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 6xy + y^2 + 32 = 0. - Colin Barker, Feb 08 2014
A6:
A7?: a(n)=(1+sqrt(2))^(2*n)+(1+sqrt(2))^(-2*n).
- Gerson Washiski Barbosa, Sep 19 2010
AUTHOR N. J. A. Sloane. No date given.
-------------------------------------------------
k = 7. x = ( 7 + sqrt(45) ) / 2 = 6.854102
2 7 47 322 2207 15127 103682
A056854
A1: a(0) = 2, a(1)= 7; for n >= 2 a(n) = 7 * a(n-1) - a(n-2).
A2: a(n) = ((7+sqrt(45))/2)^n + ((7-sqrt(45))/2)^n.
A3: G.f.: (2-7x)/(1-7x+x^2).
A4:
A5:
A6: a(n) = 7*S(n-1, 7) - 2*S(n-2, 7) = S(n, 7) - S(n-2, 7) = 2*T(n, 7/2)
AUTHOR Barry E. Williams, Aug 29, 2000.
-------------------------------------------------
k = 8. x = ( 8 + sqrt(60) ) / 2 = 7.8729833
2 8 62 488 3842 30248 238142
A086903
A1: a(0) = 2, a(1) = 8; for n >= 2 a(n) = 8 * a(n-1) - a(n-2).
A2: a(n) = (4+sqrt(15))^n + (4-sqrt(15))^n.
A3: G.f.: (2-8*x)/(1-8*x+x^2). [Philippe Deléham, Nov 02 2008]
A4:
A5: Except for the first term, positive values of x (or y)
satisfying x^2 - 8xy + y^2 + 60 = 0. - Colin Barker, Feb 13 2014
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Sep 21, 2003
-------------------------------------------------
k = 9. x = ( 9 + sqrt(77) ) / 2 = 8.8874822
2 9 79 702 6239 55449 492802
A056918
A1: a(n) = 9*a(n-1)-a(n-2); a(0)=2, a(1)=9.
A2: a(n) = ap^n + am^n, with ap := (9+sqrt(77))/2 and am := (9-sqrt(77))/2.
A3: G.f.: (2-9*x)/(1-9*x+x^2).
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 9xy + y^2 + 77 = 0. - Colin Barker, Feb 13 2014
A6: a(n) = 9*S(n-1, 9) - 2*S(n-2, 9) = S(n, 9) - S(n-2, 9) = 2*T(n, 9/2)
AUTHOR Barry E. Williams, Aug 21, 2000.
-------------------------------------------------
k = 10. x = ( 10 + sqrt(96) ) / 2 = 9.8989795
2 10 98 970 9602 95050 940898
A087799
A1: a(n) = 10*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10.
A2: a(n) = (5+sqrt(24))^n + (5-sqrt(24))^n.
A3: G.f.: (2-10*x)/(1-10*x+x^2). - Philippe Deléham, Nov 02 2008
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 10xy + y^2 + 96 = 0. - Colin Barker, Feb 25 2014
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11, 2003
-------------------------------------------------
k = 11. x = ( 11 + sqrt(117) ) / 2 = 10.908327
2 11 119 1298 14159 154451 1684802
A057076
A1: a(0) = 2, a(1) = 11; for n >= 2 a(n) = 11 * a(n-1) - a(n-2).
A2:
A3: G.f.: (2-11x)/(1-11x+x^2).
A4:
A5:
A6: a(n) = S(n, 11) - S(n-2, 11) = 2*T(n, 11/2) with S(n, x) := U(n, x/2)
AUTHOR Wolfdieter Lang, Oct 31, 2002.
-------------------------------------------------
k = 12. x = ( 12 + sqrt(140) ) / 2 = 11.91608
2 12 142 1692 20162 240252 2862862
A087800
A1: a(n) = 12*a(n-1) - a(n-2), with a(0) = 2 and a(1) = 12.
A2:
A3: G.f.: (2-12x)/(1-12x+x^2). - From Philippe Deléham, Nov 17 2008
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 12xy + y^2 + 140 = 0. - Colin Barker, Feb 25 2014
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11, 2003.
-------------------------------------------------
k = 13. x = ( 13 + sqrt(165) ) / 2 = 12.922516
2 13 167 2158 27887 360373 4656962
A078363
A1: a(n) = 13*a(n-1) - a(n-2), with a(0) = 2 and a(1) = 13.
A2: a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and
am := (13-sqrt(165))/2.
A3: G.f.: (2-13*x)/(1-13*x+x^2).
A4:
A5: Solves for x in x^2 - 3*y^2 = 4.
A6: a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2)
AUTHOR Wolfdieter Lang, Nov 29, 2002.
-------------------------------------------------
k = 14. x = ( 14 + sqrt(192) ) / 2 = 13.928203
2 14 194 2702 37634
A067902
A1: a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14.
A2: a(n) = p^n + q^n, where p = 7 + 4*sqrt(3) and
q = 7 - 4*sqrt(3). - Tanya Khovanova, Feb 06 2007
A3: G.f.: 2*(1-7*x)/(1-14*x+x^2). - N. J. A. Sloane, Nov 22 2006
A4:
A5:
A6:
AUTHOR Lekraj Beedassy, May 13, 2003.
-------------------------------------------------
k = 15. x = ( 15 + sqrt(221) ) / 2 = 14.933034
2 15 223 3330 49727 742575 11088898
A078365
A1: a(n) = 15*a(n-1) - a(n-2); a(0) = 2, a(1) = 15.
A2: a(n) = ap^n + am^n, with ap := (15+sqrt(221))/2
and am := (15-sqrt(221))/2.
A3: G.f.: (2-15*x)/(1-15*x+x^2).
A4:
A5:
A6: a(n) = S(n, 15) - S(n-2, 15) = 2*T(n, 15/2) with S(n, x) := U(n, x/2)
AUTHOR Wolfdieter Lang, Nov 29, 2002.
-------------------------------------------------
k = 16. x = ( 16 + sqrt(252) ) / 2 = 14.937254
2 16 254 4048 64514 1028176 16386302
A090727
A1: a(n) = 16*a(n-1) - a(n-2); a(0) = 2, a(1) = 16.
A2: a(n) = (8+sqrt(63))^n + (8-sqrt(63))^n.
A3: G.f.: (2-16*x)/(1-16*x+x^2). - Philippe Deléham, Nov 02 2008
A4:
A5:
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 17. x = ( 17 + sqrt(285) ) / 2 = 16.940972
2 17 287 4862 82367 1395377 23639042
A078367
A1: a(n) = 17*a(n-1) - a(n-2); a(0) = 2, a(1) = 17
A2: a(n) = (8+sqrt(63))^n + (8-sqrt(63))^n.
A3: G.f.: (2-17*x)/(1-17*x+x^2).
A4:
A5:
A6: a(n) = S(n, 17) - S(n-2, 17) = 2*T(n, 17/2) with S(n, x) := U(n, x/2)
AUTHOR Wolfdieter Lang, Nov 29, 2002.
-------------------------------------------------
k = 18. x = ( 18 + sqrt(320) ) / 2 = 17.944272
2 18 322 5778 103682 1860498 33385282
A087215
A1: a(0) = 2, a(1) = 18; for n >= 2 a(n) = 18 * a(n-1) - a(n-2).
A2: a(n) = (9 + sqrt(80))^n + (9 - sqrt(80))^n.
A3: G.f.: 2*(1-9*x)/(1-18*x+x^2). - Philippe Deléham, Nov 17 2008
A4:
A5:
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 19, 2003.
-------------------------------------------------
k = 19. x = ( 19 + sqrt(357) ) / 2 = 18.947222
2 19 359 6802 128879 2441899 46267202
A078369
A1: a(0) = 2, a(1) = 19; for n >= 2 a(n) = 19 * a(n-1) - a(n-2).
A2: a(n) = ap^n + am^n, with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2.
A3: G.f.: (2-19*x)/(1-19*x+x^2).
A4:
A5:
A6: a(n) = S(n, 19) - S(n-2, 19) = 2*T(n, 19/2) with S(n, x) := U(n, x/2)
AUTHOR Wolfdieter Lang, Nov 29, 2002.
-------------------------------------------------
k = 20. x = ( 20 + sqrt(396) ) / 2 = 18.947222
2 20 398 7940 158402 3160100 63043598
A090728
A1: a(0) = 2, a(1) = 20; for n >= 2 a(n) = 20 * a(n-1) - a(n-2).
A2: a(n) = p^n + q^n, where p = 10 + 3sqrt(11) and q = 10 - 3sqrt(11). - Tanya Khovanova, Feb 06 2007
A3: G.f.: (2-20*x)/(1-20*x+x^2). [From Philippe Deléham, Nov 02 2008]
A4:
A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 20xy + y^2 + 396 = 0. - Colin Barker, Feb 28 2014
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 21. x = ( 21 + sqrt(437) ) / 2 = 20.952272
2 21 439 9198 192719 4037901 84603202
A090729
A1: a(0) = 2, a(1) = 21; for n >= 2 a(n) = 21 * a(n-1) - a(n-2).
A2: a(n) = ap^n + am^n, with ap := (21+sqrt(437))/2 and am := (21-sqrt(437))/2.
A3: G.f.: (2-21*x)/(1-21*x+x^2).
A4:
A5:
A6: a(n) = S(n, 21) - S(n-2, 21) = 2*T(n, 21/2) with S(n, x) := U(n, x/2)
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 22. x = ( 22 + sqrt(480) ) / 2 = 21.954451
2 22 482 10582 232322 5100502
A090730
A1: a(0) = 2, a(1) = 22; for n >= 2 a(n) = 22 * a(n-1) - a(n-2).
A2: a(n) = p^n + q^n, where p = 11 + 2sqrt(30) and q = 11 - 2sqrt(30).
- Tanya Khovanova, Feb 06 2007
A3: G.f.: (2-22*x)/(1-22*x+x^2). - Philippe Deléham, Nov 18 2008
A4:
A5:
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 23. x = ( 23 + sqrt(525) ) / 2 = 22.956439
2 23 527 12098 277727 6375623
A090731
A1: a(0) = 2, a(1) = 23; for n >= 2 a(n) = 23 * a(n-1) - a(n-2).
A2: a(n) = ap^n + am^n, with ap := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.
A3: G.f.: (2-23*x)/(1-23*x+x^2).
A4:
A5:
A6: a(n) = S(n, 23) - S(n-2, 23) = 2*T(n, 23/2) with S(n, x) := U(n, x/2)
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 24. x = ( 24 + sqrt(572) ) / 2 = 23.958261
2 24 574 13752 329474 7893624
A090732
A1: a(0) = 2, a(1) = 24; for n >= 2 a(n) = 24 * a(n-1) - a(n-2).
A2: a(n) = p^n + q^n, where p = 12 + sqrt(143) and q = 12 - sqrt(143).
- Tanya Khovanova, Feb 06 2007
A3: G.f.: (2-24*x)/(1-24*x+x^2). - Philippe Deléham, Nov 02 2008
A4:
A5:
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 25. x = ( 25 + sqrt(621) ) / 2 = 24.959936
2 25 623 15550 388127 9687625
A090733
A1: a(0) = 2, a(1) = 25; for n >= 2 a(n) = 25 * a(n-1) - a(n-2).
A2: a(n) = ap^n + am^n, with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
A3: G.f.: (2-25*x)/(1-25*x+x^2).
A4:
A5: (x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative
integer solutions to x^2 - 69*(3*y)^2 =+4.
A6: a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2)
with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1.
S(n, 25)=A097780(n). U-, resp. T-, are Chebyshev's polynomials
of the second, resp. first, kind. See A049310 and A053120.
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 18, 2004.
-------------------------------------------------
k = 26. x = ( 26 + sqrt(672) ) / 2 = 25.961481
2 26 674 17498 454274 11793626
A090247
A1: a(0) = 2, a(1) = 26; for n >= 2 a(n) = 26 * a(n-1) - a(n-2).
A2: a(n) = (13+sqrt(168))^n + (13-sqrt(168))^n.
A3: G.f.: (2-26*x)/(1-26*x+x^2). - Philippe Deléham, Nov 02 2008
A4:
A5:
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24, 2004.
-------------------------------------------------
k = 27. x = ( 27 + sqrt(725) ) / 2 = 26.962912
2 27 727 19602 528527 14250627
A090248
A1: a(0) = 2, a(1) = 27; for n >= 2 a(n) = 27 * a(n-1) - a(n-2).
A2: a(n) = ((5+sqrt(29))/2)^n+((5-sqrt(29))/2)^n.
A3: G.f.: (2-27*x)/(1-27*x+x^2).
A4: E.g.f. : 2*exp(5*x/2)*cosh(sqrt(29)*x/2)
A5: a(n) gives the general (nonnegative integer) solution of the
Pell equation a^2 - 29*(5*b)^2 =+4 with companion sequence
b(n)=A097781(n-1), n>=0.
A6: a(n) = S(n, 27) - S(n-2, 27) = 2*T(n, 27/2) with S(n, x) :=
U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(n).
U-, resp. T-, are Chebyshev's polynomials of the second, resp.
first, kind.
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24, 2004.
-------------------------------------------------
k = 28. x = ( 28 + sqrt(780) ) / 2 = 27.964241
2 28 782 21868 611522 17100748
A090249
A1: a(0) = 2, a(1) = 28; for n >= 2 a(n) = 28 * a(n-1) - a(n-2).
A2: a(n) = (14+sqrt(195))^n + (14-sqrt(195))^n.
A3: G.f.: (2-28*x)/(1-28*x+x^2). - Philippe Deléham, Nov 02 2008
A4:
A5:
A6:
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24, 2004.
-------------------------------------------------
k = 29. x = ( 29 + sqrt(837) ) / 2 = 28.965476
2 29 839 24302 703919 20389349
A090251
A1: a(0) = 2, a(1) = 29; for n >= 2 a(n) = 29 * a(n-1) - a(n-2).
A2: a(n) = ((29+sqrt(837))/2)^n + ((29-sqrt(837))/2)^n.
A3: G.f.: (2-29*x)/(1-29*x+x^2).
A4:
A5:
A6: a(n) = S(n, 29) - S(n-2, 29) = 2*T(n, 29/2) with S(n, x) := U(n, x/2),
S(-1, x) := 0, S(-2, x) := -1. S(n, 27)=A097781(n). U-, resp. T-, are
Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120.
AUTHOR Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24, 2004.
-------------------------------------------------
k = 30. x = ( 30 + sqrt(896) ) / 2 = 29.966630
2 30 898 26910 806402 24165150
Not on OEIS, but see A042732.
A1. a(0) = 2, a(1) = 30; for n >= 2 a(n) = 30 * a(n-1) - a(n-2)
-------------------------------------------------
k = 31. x = ( 31 + sqrt(957) ) / 2 = 30.967708
2 31 959 29698 919679 28480351
Not on OEIS, but see A042852.
A1: a(0) = 2, a(1) = 31; for n >= 2 a(n) = 31 * a(n-1) - a(n-2)
-------------------------------------------------
k = 32. x = ( 32 + sqrt(1020) ) / 2 = 31.968719
2 32 1022 32672 1044482 33390752
Not on OEIS.
A1. a(0) = 2, a(1) = 32; for n >= 2 a(n) = 32 * a(n-1) - a(n-2)?
=========================================================================
========================================================================= p. 42
If we extract all the entries for A3, we get this list.
k = 3. A3: G.f.: (2-3*x)/(1-3*x+x^2). - Simon Plouffe in his 1992 dissertation.
k = 4. A3: G.f.: -2*(-1+2*x)/(1-4*x+x^2).
k = 5. A3: G.f.: (2-5*x)/(1-5*x+x^2). - Simon Plouffe in his 1992 dissertation.
k = 6. A3: G.f.: (2-6*x)/(1-6*x+x^2).
k = 7. A3: G.f.: (2-7*x)/(1-7*x+x^2).
k = 8. A3: G.f.: (2-8*x)/(1-8*x+x^2). [Philippe Deléham, Nov 02 2008]
k = 9. A3: G.f.: (2-9*x)/(1-9*x+x^2).
k = 10. A3: G.f.: (2-10*x)/(1-10*x+x^2). - Philippe Deléham, Nov 02 2008
k = 11. A3: G.f.: (2-11*x)/(1-11*x+x^2).
k = 12. A3: G.f.: (2-12*x)/(1-12*x+x^2). - From Philippe Deléham, Nov 17 2008
k = 13. A3: G.f.: (2-13*x)/(1-13*x+x^2).
k = 14. A3: G.f.: (2-14*x)/(1-14*x+x^2). - N. J. A. Sloane, Nov 22 2006
k = 15. A3: G.f.: (2-15*x)/(1-15*x+x^2).
k = 16. A3: G.f.: (2-16*x)/(1-16*x+x^2). - Philippe Deléham, Nov 02 2008
k = 17. A3: G.f.: (2-17*x)/(1-17*x+x^2).
k = 18. A3: G.f.: (2-18*x)/(1-18*x+x^2). - Philippe Deléham, Nov 17 2008
k = 19. A3: G.f.: (2-19*x)/(1-19*x+x^2).
k = 20. A3: G.f.: (2-20*x)/(1-20*x+x^2). [From Philippe Deléham, Nov 02 2008]
k = 21. A3: G.f.: (2-21*x)/(1-21*x+x^2).
k = 22. A3: G.f.: (2-22*x)/(1-22*x+x^2). - Philippe Deléham, Nov 18 2008
k = 23. A3: G.f.: (2-23*x)/(1-23*x+x^2).
k = 24. A3: G.f.: (2-24*x)/(1-24*x+x^2). - Philippe Deléham, Nov 02 2008
k = 25. A3: G.f.: (2-25*x)/(1-25*x+x^2).
k = 26. A3: G.f.: (2-26*x)/(1-26*x+x^2). - Philippe Deléham, Nov 02 2008
k = 27. A3: G.f.: (2-27*x)/(1-27*x+x^2).
k = 28. A3: G.f.: (2-28*x)/(1-28*x+x^2). - Philippe Deléham, Nov 02 2008
k = 29. A3: G.f.: (2-29*x)/(1-29*x+x^2).
Example. Line 5 in Table 5_14_18 is:
2 5 23 110 527 2525 12098
The A3 entry above for line 5 is:
k = 5. A3: G.f.: (2-5*x)/(1-5*x+x^2). - Simon Plouffe in his 1992 dissertation.
Doing the indicated division, we get:
2 + 5 + 23 + 110 + 527 + 2525 + ...
----------------------------------------
1 - 5 + 1 2 - 5 + 0 + 0 + 0
2 - 10 + 2
-------------------
5 - 2
5 - 25 + 5
-----------
23 - 5
23 - 115 + 23
--------------------
110 - 23
110 - 550 + 110
--------------------
527 - 110
527 - 2635 + 527
------------------
2525 - 527, etc
Based on just one example we offer the following
Conjecture. The values in line i of Table 5_14_18 are generated
by (2 - i*x)/(1 - i*x + x^2).
Hint at a proof: the mechanics of generating A1 values closely
match the mechanics of generating A3 values.
==========================================================
Such a joy to see the integers dance to the melodies played by the algorithms.
========================================================== p. 43
The Exponential Generating Function occurs often in OEIS
entries for lines in other tables, but not so frequently in this
one. Why?
k = 3. A4:
k = 4. A4: E.g.f.: 2*exp(2*x)*cosh(sqrt(3)*x). - Ilya Gutkovskiy, Apr 27 2016
k = 5. A4:
k = 6. A4:
k = 7. A4:
k = 8. A4:
k = 9. A4:
k = 10. A4:
k = 11. A4:
k = 12. A4:
k = 13. A4:
k = 14. A4:
k = 15. A4:
k = 16. A4:
k = 17. A4:
k = 18. A4:
k = 19. A4:
k = 20. A4:
k = 21. A4:
k = 22. A4:
k = 23. A4:
k = 24. A4:
k = 25. A4:
k = 26. A4:
k = 27. A4: E.g.f. : 2*exp(5*x/2)*cosh(sqrt(29)*x/2)
k = 28. A4:
k = 29. A4:
========================================================== p. 44
Here is a list of the A5 notes from the OEIS entries for the
rows of Table 5_14_18:
k = 3. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 7xy + y^2 + 45 = 0. - Colin Barker, Feb 16 2014
k = 3. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 18xy + y^2 + 320 = 0. - Colin Barker, Feb 16 2014
k = 4. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 4xy + y^2 + 12 = 0. - Colin Barker, Feb 04 2014
k = 4. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 14xy + y^2 + 192 = 0. - Colin Barker, Feb 16 2014
k = 5. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 5xy + y^2 + 21 = 0. - Colin Barker, Feb 08 2014
k = 6. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 6xy + y^2 + 32 = 0. - Colin Barker, Feb 08 2014
k = 7. A5:
k = 8. A5: Except for the first term, positive values of x (or y)
satisfying x^2 - 8xy + y^2 + 60 = 0. - Colin Barker, Feb 13 2014
k = 9. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 9xy + y^2 + 77 = 0. - Colin Barker, Feb 13 2014
k = 10. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 10xy + y^2 + 96 = 0. - Colin Barker, Feb 25 2014
k = 11. A5:
k = 12. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 12xy + y^2 + 140 = 0. - Colin Barker, Feb 25 2014
k = 13. A5: Solves for x in x^2 - 3*y^2 = 4.
k = 14. A5:
k = 15. A5:
k = 16. A5:
k = 17. A5:
k = 18. A5:
k = 19. A5:
k = 20. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 20xy + y^2 + 396 = 0. - Colin Barker, Feb 28 2014
k = 21. A5:
k = 22. A5:
k = 23. A5:
k = 24. A5:
k = 25. A5: (x,y) =(2,0), (25;1), (623;25), (15550;624), ... give the nonnegative
integer solutions to x^2 - 69*(3*y)^2 =+4.
k = 26. A5:
k = 27. A5: k = 2_. A(n) gives the general (nonnegative integer) solution of the
Pell equation a^2 - 29*(5*b)^2 =+4 with companion sequence
b(n)=A097781(n-1), n>=0.
k = 28. A5:
k = 29. A5:
OEIS was brilliant in helping at this point.
Look at the entries for lines 8, 9, and 10:
k = 8. A5: Except for the first term, positive values of x (or y)
satisfying x^2 - 8xy + y^2 + 60 = 0. - Colin Barker, Feb 13 2014
k = 9. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 9xy + y^2 + 77 = 0. - Colin Barker, Feb 13 2014
k = 10. A5: Except for the first term, positive values of x (or y) satisfying
x^2 - 10xy + y^2 + 96 = 0. - Colin Barker, Feb 25 2014
The equations
x^2 - 8xy + y^2 + 60 = 0
x^2 - 9xy + y^2 + 77 = 0
x^2 - 10xy + y^2 + 96 = 0
led to a search on OEIS for "60, 77, 96", and this led to
A028347 a(n) = n^2 - 4
0, 5, 12, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357,
396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152,
1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205,
2300, 2397, 2496, 2597
Conjecture. In Table 5_14_18 row i is generated by the
solutions to the equation x^2 - i*x*y + y^2 + A028347[i-2] = 0.
Question:
Why do some of the numbers in the A5 lines differ from the
numbers in A028347?
A nit: if a(0) = 0, then a(n) = (n+2)^2 - 4.
========================================================== p. 45
Explanation of "x is a solution of x^2 - 5xy + y^2 + 17 = 0."
A237255 Values of x in the solutions to x^2 - 5xy + y^2 + 17 = 0, where 0 < x < y.
2, 3, 7, 13, 33, 62, 158, 297, 757, 1423, 3627, 6818, 17378, 32667, 83263, 156517,
398937, 749918, 1911422, 3593073, 9158173, 17215447, 43879443, 82484162, 210239042,
395205363, 1007315767, 1893542653, 4826339793, 9072507902, 23124383198, 43468996857
COMMENTS The corresponding values of y are given by a(n+2).
EXAMPLE: 3 is in the sequence because (x, y) = (3, 13)
is a solution to x^2 - 5xy + y^2 + 17 = 0.
========================================================== p. 46
There is a clear pattern for the odd numbered lines in Table
5_14_18. I did not take the time to dig out the definitions of
S() and T().
k = 3. A6: a(n) = S(n, 3) - S(n-2, 3) = 2*T(n, 3/2) with S(n-1, 3)
k = 4. A6:
k = 5. A6: a(n) = 5*S(n-1, 5) - 2*S(n-2, 5) = S(n, 5) - S(n-2, 5) = 2*T(n, 5/2)
k = 6. A6:
k = 7. A6: A(n) = 7*S(n-1, 7) - 2*S(n-2, 7) = S(n, 7) - S(n-2, 7) = 2*T(n, 7/2)
k = 8. A6:
k = 9. A6: a(n) = 9*S(n-1, 9) - 2*S(n-2, 9) = S(n, 9) - S(n-2, 9) = 2*T(n, 9/2)
k = 10. A6:
k = 11. A6: a(n) = S(n, 11) - S(n-2, 11) = 2*T(n, 11/2) with S(n, x) := U(n, x/2)
k = 12. A6:
k = 13. A6: a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2)
k = 14. A6:
k = 15. A6: a(n) = S(n, 15) - S(n-2, 15) = 2*T(n, 15/2) with S(n, x) := U(n, x/2)
k = 16. A6:
k = 17. A6: a(n) = S(n, 17) - S(n-2, 17) = 2*T(n, 17/2) with S(n, x) := U(n, x/2)
k = 18. A6:
k = 19. A6: a(n) = S(n, 19) - S(n-2, 19) = 2*T(n, 19/2) with S(n, x) := U(n, x/2)
k = 20. A6:
k = 21. A6: a(n) = S(n, 21) - S(n-2, 21) = 2*T(n, 21/2) with S(n, x) := U(n, x/2)
k = 22. A6:
k = 23. A6: a(n) = S(n, 23) - S(n-2, 23) = 2*T(n, 23/2) with S(n, x) := U(n, x/2)
k = 24. A6:
k = 25. A6: a(n) = S(n, 25) - S(n-2, 25) = 2*T(n, 25/2)
k = 26. A6:
k = 27. A6: a(n) = S(n, 27) - S(n-2, 27) = 2*T(n, 27/2)
k = 28. A6:
k = 29. A6: a(n) = S(n, 29) - S(n-2, 29) = 2*T(n, 29/2)
Conjecture. If n is odd, then a(i,n) = S(n, i) - S(n-2, i).
========================================================== p. 47
Conclusion to Section C.
Table 5_14_18 is generated by:
A0 rule
A1 rule
A2 rule
A3 rule
A4 rule (maybe; very little evidence here)
A5 rule (some uncertainty, but lots of evidence)
A6 rule (strong evidence for the odd numbered rows)
==========================================================
========================================================== p. 48
Section D. More Tables
Table 5_14_18 came from the even diagonals of the (1,2)-Pascal triangle
Let us look at tables built from (1) all of the diagonals, (2) the even diagonals,
and (3) the odd diagonals.
The (1,2) Pascal triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+---------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 2 0 0 0 0 0 0 0 0
2| 1 3 2 0 0 0 0 0 0 0
3| 1 4 5 2 0 0 0 0 0 0
4| 1 5 9 7 2 0 0 0 0 0
5| 1 6 14 16 9 2 0 0 0 0
6| 1 7 20 30 25 11 2 0 0 0
7| 1 8 27 50 55 36 13 2 0 0
8| 1 9 35 77 105 91 49 15 2 0
9| 1 10 44 112 182 196 140 64 17 2
10| 1 11 54 156 294 378 336 204 81 19
11| 1 12 65 210 450 672 714 540 285 100
12| 1 13 77 275 660 1122 1386 1254 825 385
13| 1 14 90 352 935 1782 2508 2640 2079 1210
14| 1 15 104 442 1287 2717 4290 5148 4719 3289
15| 1 16 119 546 1729 4004 7007 9438 9867 8008
16| 1 17 135 665 2275 5733 11011 16445 19305 17875
17| 1 18 152 800 2940 8008 16744 27456 35750 37180
18| 1 19 170 952 3740 10948 24752 44200 63206 72930
19| 1 20 189 1122 4692 14688 35700 68952 107406 136136
All of the diagonals from the (1, 2) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 0 0 0 0 0 0 0 0 0
2| 1 2 0 0 0 0 0 0 0 0
3| 1 3 0 0 0 0 0 0 0 0
4| 1 4 2 0 0 0 0 0 0 0
5| 1 5 5 0 0 0 0 0 0 0
6| 1 6 9 2 0 0 0 0 0 0
7| 1 7 14 7 0 0 0 0 0 0
8| 1 8 20 16 2 0 0 0 0 0
9| 1 9 27 30 9 0 0 0 0 0
10| 1 10 35 50 25 2 0 0 0 0
11| 1 11 44 77 55 11 0 0 0 0
12| 1 12 54 112 105 36 2 0 0 0
13| 1 13 65 156 182 91 13 0 0 0
14| 1 14 77 210 294 196 49 2 0 0
15| 1 15 90 275 450 378 140 15 0 0
16| 1 16 104 352 660 672 336 64 2 0
17| 1 17 119 442 935 1122 714 204 17 0
18| 1 18 135 546 1287 1782 1386 540 81 2
19| 1 19 152 665 1729 2717 2508 1254 285 19
Table 1_4_4. All of the diagonals from the (1,2) Pascal Triangle. Source 1,2,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-------------+-+-----------------------------------------
2 |0| 2 1 2 3 2 5 2 7 2 9
1 |1| 2 1 3 4 7 11 18 29 47 76
1 2 |2| 2 1 4 5 14 19 52 71 194 265
1 3 |3| 2 1 5 6 23 29 110 139 527 666
1 4 2 |4| 2 1 6 7 34 41 198 239 1154 1393
1 5 5 |5| 2 1 7 8 47 55 322 377 2207 2584
1 6 9 2 |6| 2 1 8 9 62 71 488 559 3842 4401
1 7 14 7 |7| 2 1 9 10 79 89 702 791 6239 7030
1 8 20 16 2 |8| 2 1 10 11 98 109 970 1079 9602 10681
1 9 27 30 9 |9| 2 1 11 12 119 131 1298 1429 14159 15588
A0: p[0](n) = 2, p[1](n) = 1, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0.
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 2, a(i,1) = 1, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
1 2 1 See the next table for an
1 3 1 2 explanation of these diagrams.
--------- -----
1 4 2 1 3
1 5 5 1 4 2
------------- ---------
1 6 9 2 1 5 5
1 7 14 7 1 6 9 2
----------------- -------------
1 8 20 16 2 1 7 14 7
Table 5_14_18. The even diagonals from the (1,2) Pascal Triangle. Source 1,2,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------------------------------+-+------------------------------------------------------------------
2 |0| 2 2 2 2 2 2 2 2 2 2
1 2 |1| 2 3 7 18 47 123 322 843 2207 5778
1 4 2 |2| 2 4 14 52 194 724 2702 10084 37634 140452
1 6 9 2 |3| 2 5 23 110 527 2525 12098 57965 277727 1330670
1 8 20 16 2 |4| 2 6 34 198 1154 6726 39202 228486 1331714 7761798
1 10 35 50 25 2 |5| 2 7 47 322 2207 15127 103682 710647 4870847 33385282
1 12 54 112 105 36 2 |6| 2 8 62 488 3842 30248 238142 1874888 14760962 116212808
1 14 77 210 294 196 49 2 |7| 2 9 79 702 6239 55449 492802 4379769 38925119 345946302
1 16 104 352 660 672 336 64 2 |8| 2 10 98 970 9602 95050 940898 9313930 92198402 912670090
1 18 135 546 1287 1782 1386 540 81 2 |9| 2 11 119 1298 14159 154451 1684802 18378371 200477279 2186871698
A0: p[0](n) = 2, p[1](n) = n+2, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0.
A1: a(i,0) = 2, a(i,1) = i+2, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
Diagrams to assist in verifying the values in the
left hand side of the above table:
The diagram on the right shows the calculations
p[3](n) = (n+2) * p[2](n) - p[1](n), and
p[4](n) = (n+2) * p[3](n) - p[2](n).
The diagram on the left shows the same calculations
with everything except the coefficients removed.
In subsequent examples just the left hand diagram is shown.
(n + 2) * (n^2 + 4*n + 2) - (n + 2) =
1 4 2 n^3 + 4*n^2 + 2*n
+ 2 8 4 + 2*n^2 + 8*n + 4
------------- ---------------------
1 6 10 4 n^3 + 6*n^2 + 10*n + 4
- 1 2 - n + 2
------------- ----------------------
1 6 9 2 n^3 + 6*n^2 + 9*n + 2
(n + 2) * (n^3 + 6*n^2 + 9*n + 2) - (n^2 + 4*n + 2) =
1 6 9 2 n^4 + 6*n^3 + 9*n^2 + 2*n
+ 2 12 18 4 + 2*n^3 + 12*n^2 + 18*n + 4
----------------- --------------------------------
1 8 21 20 4 n^4 + 8*n^3 + 21*n^2 + 20*n + 4
- 1 4 2 - n^2 + 4*n + 2
----------------- --------------------------------
1 8 20 16 2 n^4 + 8*n^3 + 20*n^2 + 16*n + 2
Table 6_19_29. The odd diagonals from the (1,2) Pascal Triangle. Source 1,2,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-----------------------------------------+-+------------------------------------------------------------------
1 |0| 1 3 5 7 9 11 13 15 17 19
1 3 |1| 1 4 11 29 76 199 521 1364 3571 9349
1 5 5 |2| 1 5 19 71 265 989 3691 13775 51409 191861
1 7 14 7 |3| 1 6 29 139 666 3191 15289 73254 350981 1681651
1 9 27 30 9 |4| 1 7 41 239 1393 8119 47321 275807 1607521 9369319
1 11 44 77 55 11 |5| 1 8 55 377 2584 17711 121393 832040 5702887 39088169
1 13 65 156 182 91 13 |6| 1 9 71 559 4401 34649 272791 2147679 16908641 133121449
1 15 90 275 450 378 140 15 |7| 1 10 89 791 7030 62479 555281 4935050 43860169 389806471
1 17 119 442 935 1122 714 204 17 |8| 1 11 109 1079 10681 105731 1046629 10360559 102558961 1015229051
1 19 152 665 1729 2717 2508 1254 285 19 |9| 1 12 131 1429 15588 170039 1854841 20233212 220710491 2407582189
A0: p[0](n) = 1, p[1](n) = n+3, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0.
A1: a(i,0) = 1, a(i,1) = i+3, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 5 5
+ 2 10 10
-------------
1 7 15 10
- 1 3
-------------
1 7 14 7
+ 2 14 28 14
-----------------
1 9 28 35 14
- 1 5 5
-----------------
1 9 27 30 9
We can use the (a,b) Pascal triangles to generate many more wisteria tables.
From the (1,1) Pascal Triangle.
Table 1_3_3. All of the diagonals. Source 1,1,0,1.
Table 4_11_13. The even diagonals. Source 1,1,0,2.
Table 5_15_21. The odd diagonals. Source 1,1,1,2.
From the (1,2) Pascal Triangle.
Table 1_4_4. All of the diagonals. Source 1,2,0,1.
Table 5_14_18. The even diagonals. Source 1,2,0,2.
Table 6_19_29. The odd diagonals. Source 1,2,1,2.
From the (2,1) Pascal Triangle.
Table 2_5_5_7_7. All of the diagonals. Source 2,1,0,1.
Table 7_19_21. The even diagonals. Source 2,1,0,2.
Table 9_26_34. The odd diagonals. Source 2,1,1,2.
From the (1,3) Pascal Triangle.
Table 1_5_5. All of the diagonals. Source 1,3,0,1.
Table 6_17_23. The even diagonals. Source 1,3,0,2.
Table 7_23_37. The odd diagonals. Source 1,3,1,2.
From the (3,1) Pascal Triangle.
Table 3_7_7. All of the diagonals. Source 3,1,0,1.
Table 10_27_29. The even diagonals. Source 3,1,0,2.
Table 13_37_47. The odd diagonals. Source 3,1,1,2.
From the (1,4) Pascal Triangle.
Table 1_6_6. All of the diagonals. Source 1,4,0,1.
Table 7_20_28. The even diagonals. Source 1,4,0,2.
Table 8_27_45. The odd diagonals. Source 1,4,1,2.
From the (2,3) Pascal Triangle.
Table 2_7_7. All of the diagonals. Source 2,3,0,1.
Table 9_25_31. The even diagonals. Source 2,3,0,2.
Table 11_34_50. The odd diagonals. Source 2,3,1,2.
From the (3,2) Pascal Triangle.
Table 3_8_8. All of the diagonals. Source 3,2,0,1.
Table 11_30_34. The even diagonals. Source 3,2,0,2.
Table 14_41_55. The odd diagonals. Source 3,2,1,2.
From the (4,1) Pascal Triangle.
Table 4_9_9. All of the diagonals. Source 4,1,0,1.
Table 13_35_37. The even diagonals. Source 4,1,0,2.
Table 17_48_60. The odd diagonals. Source 4,1,1,2.
From the (3,7) Pascal Triangle.
Table 3_13_13. All of the diagonals. Source 3,7,0,1.
Table 16_45_59. The even diagonals. Source 3,7,0,2.
Table 19_61_95. The odd diagonals. Source 3,7,1,2.
Miscellaneous tables.
Table 9_33_67.
Table 2_5_5_7_12.
Table 2_4_5.
Table 1_5_4.
Table 2_3_-1.
"Source (a,b,c,d)" identifies the source of the table as being
from the (a,b)-Pascal Triangle, using diagonals c (mod d), that
is, (a,b,0,1) means all diagonals, (a,b,0,2) means the even
diagonals, and (a,b,1,2) means the odd diagonals.
For tables not originating from a Pascal triangle, the Parms
parameter is another way of describing the table.
Parms (a b, c d, e f, g h) says that the A0 algorithm for the
table is:
A0: p[0](n) = e*n+f, p[1](n) = g*n+h, n>=0; p[j](n) = (a*n+b) * p[j-1](n) + (c*n+d) * p[j-2](n), j>=2,n>=0.
If a, c, e, or g is zero, it is omitted.
Are there wisteria tables which do not originate in a
Pascal triangle? The answer is not clear.
To see the A0 and A1 statements for tables based on all
of the diagonals in Pascal tables, search for "zz0".
To see the A0 and A1 statements for tables based on all of the
even numbered diagonals and the odd numbered diagonals in Pascal
tables, search for "zz1".
To see the A0 and A1 statements for other tables search for "zz*".
============================================================================================================
= =
= Enter build3(1,1) =
= =
============================================================================================================
The (1, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 1 0 0 0 0 0 0 0 0
2| 1 2 1 0 0 0 0 0 0 0
3| 1 3 3 1 0 0 0 0 0 0
4| 1 4 6 4 1 0 0 0 0 0
5| 1 5 10 10 5 1 0 0 0 0
6| 1 6 15 20 15 6 1 0 0 0
7| 1 7 21 35 35 21 7 1 0 0
8| 1 8 28 56 70 56 28 8 1 0
9| 1 9 36 84 126 126 84 36 9 1
10| 1 10 45 120 210 252 210 120 45 10
11| 1 11 55 165 330 462 462 330 165 55
12| 1 12 66 220 495 792 924 792 495 220
13| 1 13 78 286 715 1287 1716 1716 1287 715
14| 1 14 91 364 1001 2002 3003 3432 3003 2002
15| 1 15 105 455 1365 3003 5005 6435 6435 5005
16| 1 16 120 560 1820 4368 8008 11440 12870 11440
17| 1 17 136 680 2380 6188 12376 19448 24310 24310
18| 1 18 153 816 3060 8568 18564 31824 43758 48620
19| 1 19 171 969 3876 11628 27132 50388 75582 92378
All of the diagonals from the (1, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+---------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 0 0 0 0 0 0 0 0 0
2| 1 1 0 0 0 0 0 0 0 0
3| 1 2 0 0 0 0 0 0 0 0
4| 1 3 1 0 0 0 0 0 0 0
5| 1 4 3 0 0 0 0 0 0 0
6| 1 5 6 1 0 0 0 0 0 0
7| 1 6 10 4 0 0 0 0 0 0
8| 1 7 15 10 1 0 0 0 0 0
9| 1 8 21 20 5 0 0 0 0 0
10| 1 9 28 35 15 1 0 0 0 0
11| 1 10 36 56 35 6 0 0 0 0
12| 1 11 45 84 70 21 1 0 0 0
13| 1 12 55 120 126 56 7 0 0 0
14| 1 13 66 165 210 126 28 1 0 0
15| 1 14 78 220 330 252 84 8 0 0
16| 1 15 91 286 495 462 210 36 1 0
17| 1 16 105 364 715 792 462 120 9 0
18| 1 17 120 455 1001 1287 924 330 45 1
19| 1 18 136 560 1365 2002 1716 792 165 10
Table 1_3_3.. All of the diagonals from the (1,1) Pascal Triangle. Source 1,1,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-------------+-+-----------------------------------------
1 |0| 1 1 1 2 1 3 1 4 1 5
1 |1| 1 1 2 3 5 8 13 21 34 55
1 1 |2| 1 1 3 4 11 15 41 56 153 209
1 2 |3| 1 1 4 5 19 24 91 115 436 551
1 3 1 |4| 1 1 5 6 29 35 169 204 985 1189
1 4 3 |5| 1 1 6 7 41 48 281 329 1926 2255
1 5 6 1 |6| 1 1 7 8 55 63 433 496 3409 3905
1 6 10 4 |7| 1 1 8 9 71 80 631 711 5608 6319
1 7 15 10 1 |8| 1 1 9 10 89 99 881 980 8721 9701
1 8 21 20 5 |9| 1 1 10 11 109 120 1189 1309 12970 14279
A0: p[0](n) = 1, p[1](n) = 1, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 1, a(i,1) = 1, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
1 1 1
+ 1 2 + 1 1
--------- -----
1 3 1 1 2
+ 1 4 3 + 1 3 1
------------- ---------
1 5 6 1 1 4 3
+ 1 6 10 4 + 1 5 6 1
----------------- -------------
1 7 15 10 1 1 6 10 4
Table 4_11_13.. The even diagonals from the (1,1) Pascal Triangle. Source 1,1,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-------------------------------------+-+------------------------------------------------------------------
1 |0| 1 1 1 1 1 1 1 1 1 1
1 1 |1| 1 2 5 13 34 89 233 610 1597 4181
1 3 1 |2| 1 3 11 41 153 571 2131 7953 29681 110771
1 5 6 1 |3| 1 4 19 91 436 2089 10009 47956 229771 1100899
1 7 15 10 1 |4| 1 5 29 169 985 5741 33461 195025 1136689 6625109
1 9 28 35 15 1 |5| 1 6 41 281 1926 13201 90481 620166 4250681 29134601
1 11 45 84 70 21 1 |6| 1 7 55 433 3409 26839 211303 1663585 13097377 103115431
1 13 66 165 210 126 28 1 |7| 1 8 71 631 5608 49841 442961 3936808 34988311 310957991
1 15 91 286 495 462 210 36 1 |8| 1 9 89 881 8721 86329 854569 8459361 83739041 828931049
1 17 120 455 1001 1287 924 330 45 1 |9| 1 10 109 1189 12970 141481 1543321 16835050 183642229 2003229469
A0: p[0](n) = 1, p[1](n) = n+1, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = i+1, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 3 1
+ 2 6 2
-------------
1 5 7 2
- 1 1
-------------
1 5 6 1
+ 2 10 12 2
-----------------
1 7 16 13 2
- 1 3 1
-----------------
1 7 15 10 1
Table 5_15_21. The odd diagonals from the (1,1) Pascal Triangle. Source 1,1,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
----------------------------------------+-+------------------------------------------------------------------
1 |0| 1 2 3 4 5 6 7 8 9 10
1 2 |1| 1 3 8 21 55 144 377 987 2584 6765
1 4 3 |2| 1 4 15 56 209 780 2911 10864 40545 151316
1 6 10 4 |3| 1 5 24 115 551 2640 12649 60605 290376 1391275
1 8 21 20 5 |4| 1 6 35 204 1189 6930 40391 235416 1372105 7997214
1 10 36 56 35 6 |5| 1 7 48 329 2255 15456 105937 726103 4976784 34111385
1 12 55 120 126 56 7 |6| 1 8 63 496 3905 30744 242047 1905632 15003009 118118440
1 14 78 220 330 252 84 8 |7| 1 9 80 711 6319 56160 499121 4435929 39424240 350382231
1 16 105 364 715 792 462 120 9 |8| 1 10 99 980 9701 96030 950599 9409960 93149001 922080050
1 18 136 560 1365 2002 1716 792 165 10 |9| 1 11 120 1309 14279 155760 1699081 18534131 202176360 2205405829
A0: p[0](n) = 1, p[1](n) = n+2, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = i+2, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 4 3
+ 2 8 6
-------------
1 6 11 6
- 1 2
-------------
1 6 10 4
+ 2 12 20 8
-----------------
1 8 22 24 8
- 1 4 3
-----------------
1 8 21 20 5
============================================================================================================
= =
= Exit build3(1,1) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(1,2) =
= =
============================================================================================================
The (1, 2) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+---------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 2 0 0 0 0 0 0 0 0
2| 1 3 2 0 0 0 0 0 0 0
3| 1 4 5 2 0 0 0 0 0 0
4| 1 5 9 7 2 0 0 0 0 0
5| 1 6 14 16 9 2 0 0 0 0
6| 1 7 20 30 25 11 2 0 0 0
7| 1 8 27 50 55 36 13 2 0 0
8| 1 9 35 77 105 91 49 15 2 0
9| 1 10 44 112 182 196 140 64 17 2
10| 1 11 54 156 294 378 336 204 81 19
11| 1 12 65 210 450 672 714 540 285 100
12| 1 13 77 275 660 1122 1386 1254 825 385
13| 1 14 90 352 935 1782 2508 2640 2079 1210
14| 1 15 104 442 1287 2717 4290 5148 4719 3289
15| 1 16 119 546 1729 4004 7007 9438 9867 8008
16| 1 17 135 665 2275 5733 11011 16445 19305 17875
17| 1 18 152 800 2940 8008 16744 27456 35750 37180
18| 1 19 170 952 3740 10948 24752 44200 63206 72930
19| 1 20 189 1122 4692 14688 35700 68952 107406 136136
All of the diagonals from the (1, 2) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 0 0 0 0 0 0 0 0 0
2| 1 2 0 0 0 0 0 0 0 0
3| 1 3 0 0 0 0 0 0 0 0
4| 1 4 2 0 0 0 0 0 0 0
5| 1 5 5 0 0 0 0 0 0 0
6| 1 6 9 2 0 0 0 0 0 0
7| 1 7 14 7 0 0 0 0 0 0
8| 1 8 20 16 2 0 0 0 0 0
9| 1 9 27 30 9 0 0 0 0 0
10| 1 10 35 50 25 2 0 0 0 0
11| 1 11 44 77 55 11 0 0 0 0
12| 1 12 54 112 105 36 2 0 0 0
13| 1 13 65 156 182 91 13 0 0 0
14| 1 14 77 210 294 196 49 2 0 0
15| 1 15 90 275 450 378 140 15 0 0
16| 1 16 104 352 660 672 336 64 2 0
17| 1 17 119 442 935 1122 714 204 17 0
18| 1 18 135 546 1287 1782 1386 540 81 2
19| 1 19 152 665 1729 2717 2508 1254 285 19
Table 1_4_4. All of the diagonals from the (1,2) Pascal Triangle. Source 1,2,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-------------+-+-----------------------------------------
2 |0| 2 1 2 3 2 5 2 7 2 9
1 |1| 2 1 3 4 7 11 18 29 47 76
1 2 |2| 2 1 4 5 14 19 52 71 194 265
1 3 |3| 2 1 5 6 23 29 110 139 527 666
1 4 2 |4| 2 1 6 7 34 41 198 239 1154 1393
1 5 5 |5| 2 1 7 8 47 55 322 377 2207 2584
1 6 9 2 |6| 2 1 8 9 62 71 488 559 3842 4401
1 7 14 7 |7| 2 1 9 10 79 89 702 791 6239 7030
1 8 20 16 2 |8| 2 1 10 11 98 109 970 1079 9602 10681
1 9 27 30 9 |9| 2 1 11 12 119 131 1298 1429 14159 15588
A0: p[0](n) = 2, p[1](n) = 1, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 2, a(i,1) = 1, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
1 2 1
+ 1 3 + 1 2
--------- -----
1 4 2 1 3
+ 1 5 5 + 1 4 2
------------- ---------
1 6 9 2 1 5 5
+ 1 7 14 7 + 1 6 9 2
----------------- -------------
1 8 20 16 2 1 7 14 7
Table 5_14_18. The even diagonals from the (1,2) Pascal Triangle. Source 1,2,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------------------------------+-+------------------------------------------------------------------
2 |0| 2 2 2 2 2 2 2 2 2 2
1 2 |1| 2 3 7 18 47 123 322 843 2207 5778
1 4 2 |2| 2 4 14 52 194 724 2702 10084 37634 140452
1 6 9 2 |3| 2 5 23 110 527 2525 12098 57965 277727 1330670
1 8 20 16 2 |4| 2 6 34 198 1154 6726 39202 228486 1331714 7761798
1 10 35 50 25 2 |5| 2 7 47 322 2207 15127 103682 710647 4870847 33385282
1 12 54 112 105 36 2 |6| 2 8 62 488 3842 30248 238142 1874888 14760962 116212808
1 14 77 210 294 196 49 2 |7| 2 9 79 702 6239 55449 492802 4379769 38925119 345946302
1 16 104 352 660 672 336 64 2 |8| 2 10 98 970 9602 95050 940898 9313930 92198402 912670090
1 18 135 546 1287 1782 1386 540 81 2 |9| 2 11 119 1298 14159 154451 1684802 18378371 200477279 2186871698
A0: p[0](n) = 2, p[1](n) = n+2, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 2, a(i,1) = i+2, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 4 2
+ 2 8 4
-------------
1 6 10 4
- 1 2
-------------
1 6 9 2
+ 2 12 18 4
-----------------
1 8 21 20 4
- 1 4 2
-----------------
1 8 20 16 2
Table 6_19_29. The odd diagonals from the (1,2) Pascal Triangle. Source 1,2,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-----------------------------------------+-+------------------------------------------------------------------
1 |0| 1 3 5 7 9 11 13 15 17 19
1 3 |1| 1 4 11 29 76 199 521 1364 3571 9349
1 5 5 |2| 1 5 19 71 265 989 3691 13775 51409 191861
1 7 14 7 |3| 1 6 29 139 666 3191 15289 73254 350981 1681651
1 9 27 30 9 |4| 1 7 41 239 1393 8119 47321 275807 1607521 9369319
1 11 44 77 55 11 |5| 1 8 55 377 2584 17711 121393 832040 5702887 39088169
1 13 65 156 182 91 13 |6| 1 9 71 559 4401 34649 272791 2147679 16908641 133121449
1 15 90 275 450 378 140 15 |7| 1 10 89 791 7030 62479 555281 4935050 43860169 389806471
1 17 119 442 935 1122 714 204 17 |8| 1 11 109 1079 10681 105731 1046629 10360559 102558961 1015229051
1 19 152 665 1729 2717 2508 1254 285 19 |9| 1 12 131 1429 15588 170039 1854841 20233212 220710491 2407582189
A0: p[0](n) = 1, p[1](n) = n+3, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = i+3, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 5 5
+ 2 10 10
-------------
1 7 15 10
- 1 3
-------------
1 7 14 7
+ 2 14 28 14
-----------------
1 9 28 35 14
- 1 5 5
-----------------
1 9 27 30 9
============================================================================================================
= =
= Exit build3(1,2) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(2,1) =
= =
============================================================================================================
The (2, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+---------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 2 1 0 0 0 0 0 0 0 0
2| 2 3 1 0 0 0 0 0 0 0
3| 2 5 4 1 0 0 0 0 0 0
4| 2 7 9 5 1 0 0 0 0 0
5| 2 9 16 14 6 1 0 0 0 0
6| 2 11 25 30 20 7 1 0 0 0
7| 2 13 36 55 50 27 8 1 0 0
8| 2 15 49 91 105 77 35 9 1 0
9| 2 17 64 140 196 182 112 44 10 1
10| 2 19 81 204 336 378 294 156 54 11
11| 2 21 100 285 540 714 672 450 210 65
12| 2 23 121 385 825 1254 1386 1122 660 275
13| 2 25 144 506 1210 2079 2640 2508 1782 935
14| 2 27 169 650 1716 3289 4719 5148 4290 2717
15| 2 29 196 819 2366 5005 8008 9867 9438 7007
16| 2 31 225 1015 3185 7371 13013 17875 19305 16445
17| 2 33 256 1240 4200 10556 20384 30888 37180 35750
18| 2 35 289 1496 5440 14756 30940 51272 68068 72930
19| 2 37 324 1785 6936 20196 45696 82212 119340 140998
All of the diagonals from the (2, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 2 0 0 0 0 0 0 0 0 0
2| 2 1 0 0 0 0 0 0 0 0
3| 2 3 0 0 0 0 0 0 0 0
4| 2 5 1 0 0 0 0 0 0 0
5| 2 7 4 0 0 0 0 0 0 0
6| 2 9 9 1 0 0 0 0 0 0
7| 2 11 16 5 0 0 0 0 0 0
8| 2 13 25 14 1 0 0 0 0 0
9| 2 15 36 30 6 0 0 0 0 0
10| 2 17 49 55 20 1 0 0 0 0
11| 2 19 64 91 50 7 0 0 0 0
12| 2 21 81 140 105 27 1 0 0 0
13| 2 23 100 204 196 77 8 0 0 0
14| 2 25 121 285 336 182 35 1 0 0
15| 2 27 144 385 540 378 112 9 0 0
16| 2 29 169 506 825 714 294 44 1 0
17| 2 31 196 650 1210 1254 672 156 10 0
18| 2 33 225 819 1716 2079 1386 450 54 1
19| 2 35 256 1015 2366 3289 2640 1122 210 11
Table 2_5_5_7_7. All of the diagonals from the (2,1) Pascal Triangle. Source 2,1,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------+-+-----------------------------------------
1 |0| 1 2 1 3 1 4 1 5 1 6
2 |1| 1 2 3 5 8 13 21 34 55 89
2 1 |2| 1 2 5 7 19 26 71 97 265 362
2 3 |3| 1 2 7 9 34 43 163 206 781 987
2 5 1 |4| 1 2 9 11 53 64 309 373 1801 2174
2 7 4 |5| 1 2 11 13 76 89 521 610 3571 4181
2 9 9 1 |6| 1 2 13 15 103 118 811 929 6385 7314
2 11 16 5 |7| 1 2 15 17 134 151 1191 1342 10585 11927
2 13 25 14 1 |8| 1 2 17 19 169 188 1673 1861 16561 18422
2 15 36 30 6 |9| 1 2 19 21 208 229 2269 2498 24751 27249
A0: p[0](n) = 1, p[1](n) = 2, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 1, a(i,1) = 2, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
2 1 2
+ 2 3 + 2 1
--------- -----
2 5 1 2 3
+ 2 7 4 + 2 5 1
------------- ---------
2 9 9 1 2 7 4
+ 2 11 16 5 + 2 9 9 1
----------------- -------------
2 13 25 14 1 2 11 16 5
Table 7_19_21. The even diagonals from the (2,1) Pascal Triangle. Source 2,1,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------------------------------+-+------------------------------------------------------------------
1 |0| 1 1 1 1 1 1 1 1 1 1
2 1 |1| 1 3 8 21 55 144 377 987 2584 6765
2 5 1 |2| 1 5 19 71 265 989 3691 13775 51409 191861
2 9 9 1 |3| 1 7 34 163 781 3742 17929 85903 411586 1972027
2 13 25 14 1 |4| 1 9 53 309 1801 10497 61181 356589 2078353 12113529
2 17 49 55 20 1 |5| 1 11 76 521 3571 24476 167761 1149851 7881196 54018521
2 21 81 140 105 27 1 |6| 1 13 103 811 6385 50269 395767 3115867 24531169 193133485
2 25 121 285 336 182 35 1 |7| 1 15 134 1191 10585 94074 836081 7430655 66039814 586927671
2 29 169 506 825 714 294 44 1 |8| 1 17 169 1673 16561 163937 1622809 16064153 159018721 1574123057
2 33 225 819 1716 2079 1386 450 54 1 |9| 1 19 208 2269 24751 269992 2945161 32126779 350449408 3822816709
A0: p[0](n) = 1, p[1](n) = 2*n+1, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = 2*i+1, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
2 5 1
+ 4 10 2
-------------
2 9 11 2
- 2 1
-------------
2 9 9 1
+ 4 18 18 2
-----------------
2 13 27 19 2
- 2 5 1
-----------------
2 13 25 14 1
Table 9_26_34. The odd diagonals from the (2,1) Pascal Triangle. Source 2,1,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------------------------+-+------------------------------------------------------------------
2 |0| 2 3 4 5 6 7 8 9 10 11
2 3 |1| 2 5 13 34 89 233 610 1597 4181 10946
2 7 4 |2| 2 7 26 97 362 1351 5042 18817 70226 262087
2 11 16 5 |3| 2 9 43 206 987 4729 22658 108561 520147 2492174
2 15 36 30 6 |4| 2 11 64 373 2174 12671 73852 430441 2508794 14622323
2 19 64 91 50 7 |5| 2 13 89 610 4181 28657 196418 1346269 9227465 63245986
2 23 100 204 196 77 8 |6| 2 15 118 929 7314 57583 453350 3569217 28100386 221233871
2 27 144 385 540 378 112 9 |7| 2 17 151 1342 11927 106001 942082 8372737 74412551 661340222
2 31 196 650 1210 1254 672 156 10 |8| 2 19 188 1861 18422 182359 1805168 17869321 176888042 1751011099
2 35 256 1015 2366 3289 2640 1122 210 11 |9| 2 21 229 2498 27249 297241 3242402 35369181 385818589 4208635298
A0: p[0](n) = 2, p[1](n) = 2*n+3, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 2, a(i,1) = 2*i+3, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
2 7 4
+ 4 14 8
-------------
2 11 18 8
- 2 3
-------------
2 11 16 5
+ 4 22 32 10
-----------------
2 15 38 37 10
- 2 7 4
-----------------
2 15 36 30 6
============================================================================================================
= =
= Exit build3(2,1) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(1,3) =
= =
============================================================================================================
The (1, 3) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+---------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 3 0 0 0 0 0 0 0 0
2| 1 4 3 0 0 0 0 0 0 0
3| 1 5 7 3 0 0 0 0 0 0
4| 1 6 12 10 3 0 0 0 0 0
5| 1 7 18 22 13 3 0 0 0 0
6| 1 8 25 40 35 16 3 0 0 0
7| 1 9 33 65 75 51 19 3 0 0
8| 1 10 42 98 140 126 70 22 3 0
9| 1 11 52 140 238 266 196 92 25 3
10| 1 12 63 192 378 504 462 288 117 28
11| 1 13 75 255 570 882 966 750 405 145
12| 1 14 88 330 825 1452 1848 1716 1155 550
13| 1 15 102 418 1155 2277 3300 3564 2871 1705
14| 1 16 117 520 1573 3432 5577 6864 6435 4576
15| 1 17 133 637 2093 5005 9009 12441 13299 11011
16| 1 18 150 770 2730 7098 14014 21450 25740 24310
17| 1 19 168 920 3500 9828 21112 35464 47190 50050
18| 1 20 187 1088 4420 13328 30940 56576 82654 97240
19| 1 21 207 1275 5508 17748 44268 87516 139230 179894
All of the diagonals from the (1, 3) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------
0| 3 0 0 0 0 0 0 0 0 0
1| 1 0 0 0 0 0 0 0 0 0
2| 1 3 0 0 0 0 0 0 0 0
3| 1 4 0 0 0 0 0 0 0 0
4| 1 5 3 0 0 0 0 0 0 0
5| 1 6 7 0 0 0 0 0 0 0
6| 1 7 12 3 0 0 0 0 0 0
7| 1 8 18 10 0 0 0 0 0 0
8| 1 9 25 22 3 0 0 0 0 0
9| 1 10 33 40 13 0 0 0 0 0
10| 1 11 42 65 35 3 0 0 0 0
11| 1 12 52 98 75 16 0 0 0 0
12| 1 13 63 140 140 51 3 0 0 0
13| 1 14 75 192 238 126 19 0 0 0
14| 1 15 88 255 378 266 70 3 0 0
15| 1 16 102 330 570 504 196 22 0 0
16| 1 17 117 418 825 882 462 92 3 0
17| 1 18 133 520 1155 1452 966 288 25 0
18| 1 19 150 637 1573 2277 1848 750 117 3
19| 1 20 168 770 2093 3432 3300 1716 405 28
Table 1_5_5. All of the diagonals from the (1,3) Pascal Triangle. Source 1,3,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------+-+-----------------------------------------
3 |0| 3 1 3 4 3 7 3 10 3 13
1 |1| 3 1 4 5 9 14 23 37 60 97
1 3 |2| 3 1 5 6 17 23 63 86 235 321
1 4 |3| 3 1 6 7 27 34 129 163 618 781
1 5 3 |4| 3 1 7 8 39 47 227 274 1323 1597
1 6 7 |5| 3 1 8 9 53 62 363 425 2488 2913
1 7 12 3 |6| 3 1 9 10 69 79 543 622 4275 4897
1 8 18 10 |7| 3 1 10 11 87 98 773 871 6870 7741
1 9 25 22 3 |8| 3 1 11 12 107 119 1059 1178 10483 11661
1 10 33 40 13 |9| 3 1 12 13 129 142 1407 1549 15348 16897
A0: p[0](n) = 3, p[1](n) = 1, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 3, a(i,1) = 1, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
1 3 1
+ 1 4 + 1 3
--------- -----
1 5 3 1 4
+ 1 6 7 + 1 5 3
------------- ---------
1 7 12 3 1 6 7
+ 1 8 18 10 + 1 7 12 3
----------------- -------------
1 9 25 22 3 1 8 18 10
Table 6_17_23. The even diagonals from the (1,3) Pascal Triangle. Source 1,3,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------------------------------+-+------------------------------------------------------------------
3 |0| 3 3 3 3 3 3 3 3 3 3
1 3 |1| 3 4 9 23 60 157 411 1076 2817 7375
1 5 3 |2| 3 5 17 63 235 877 3273 12215 45587 170133
1 7 12 3 |3| 3 6 27 129 618 2961 14187 67974 325683 1560441
1 9 25 22 3 |4| 3 7 39 227 1323 7711 44943 261947 1526739 8898487
1 11 42 65 35 3 |5| 3 8 53 363 2488 17053 116883 801128 5491013 37635963
1 13 63 140 140 51 3 |6| 3 9 69 543 4275 33657 264981 2086191 16424547 129310185
1 15 88 255 378 266 70 3 |7| 3 10 87 773 6870 61057 542643 4822730 42861927 380934613
1 17 117 418 825 882 462 92 3 |8| 3 11 107 1059 10483 103771 1027227 10168499 100657763 996409131
1 19 150 637 1573 2277 1848 750 117 3 |9| 3 12 129 1407 15348 167421 1826283 19921692 217312329 2370513927
A0: p[0](n) = 3; p[1](n) = n+3, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 3, a(i,1) = i+3, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 5 3
+ 2 10 6
-------------
1 7 13 6
- 1 3
-------------
1 7 12 3
+ 2 14 24 6
-----------------
1 9 26 27 6
- 1 5 3
-----------------
1 9 25 22 3
Table 7_23_37. The odd diagonals from the (1,3) Pascal Triangle. Source 1,3,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-----------------------------------------+-+------------------------------------------------------------------
1 |0| 1 4 7 10 13 16 19 22 25 28
1 4 |1| 1 5 14 37 97 254 665 1741 4558 11933
1 6 7 |2| 1 6 23 86 321 1198 4471 16686 62273 232406
1 8 18 10 |3| 1 7 34 163 781 3742 17929 85903 411586 1972027
1 10 33 40 13 |4| 1 8 47 274 1597 9308 54251 316198 1842937 10741424
1 12 52 98 75 16 |5| 1 9 62 425 2913 19966 136849 937977 6428990 44064953
1 14 75 192 238 126 19 |6| 1 10 79 622 4897 38554 303535 2389726 18814273 148124458
1 16 102 330 570 504 196 22 |7| 1 11 98 871 7741 68798 611441 5434171 48296098 429230711
1 18 133 520 1155 1452 966 288 25 |8| 1 12 119 1178 11661 115432 1142659 11311158 111968921 1108378052
1 20 168 770 2093 3432 3300 1716 405 28 |9| 1 13 142 1549 16897 184318 2010601 21932293 239244622 2609758549
A0: p[0](n) = 1, p[1](n) = n+4, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = i+4, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 6 7
+ 2 12 14
-------------
1 8 19 14
- 1 4
-------------
1 8 18 10
+ 2 16 36 20
-----------------
1 10 34 46 20
- 1 6 7
-----------------
1 10 33 40 13
============================================================================================================
= =
= Exit build3(1,3) =
= =
============================================================================================================
============================================================================================================
= =
= Skip build3(2,2) It is 2 times build3(1,1). =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(3,1) =
= =
============================================================================================================
The (3, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 3 1 0 0 0 0 0 0 0 0
2| 3 4 1 0 0 0 0 0 0 0
3| 3 7 5 1 0 0 0 0 0 0
4| 3 10 12 6 1 0 0 0 0 0
5| 3 13 22 18 7 1 0 0 0 0
6| 3 16 35 40 25 8 1 0 0 0
7| 3 19 51 75 65 33 9 1 0 0
8| 3 22 70 126 140 98 42 10 1 0
9| 3 25 92 196 266 238 140 52 11 1
10| 3 28 117 288 462 504 378 192 63 12
11| 3 31 145 405 750 966 882 570 255 75
12| 3 34 176 550 1155 1716 1848 1452 825 330
13| 3 37 210 726 1705 2871 3564 3300 2277 1155
14| 3 40 247 936 2431 4576 6435 6864 5577 3432
15| 3 43 287 1183 3367 7007 11011 13299 12441 9009
16| 3 46 330 1470 4550 10374 18018 24310 25740 21450
17| 3 49 376 1800 6020 14924 28392 42328 50050 47190
18| 3 52 425 2176 7820 20944 43316 70720 92378 97240
19| 3 55 477 2601 9996 28764 64260 114036 163098 189618
All of the diagonals from the (3, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 3 0 0 0 0 0 0 0 0 0
2| 3 1 0 0 0 0 0 0 0 0
3| 3 4 0 0 0 0 0 0 0 0
4| 3 7 1 0 0 0 0 0 0 0
5| 3 10 5 0 0 0 0 0 0 0
6| 3 13 12 1 0 0 0 0 0 0
7| 3 16 22 6 0 0 0 0 0 0
8| 3 19 35 18 1 0 0 0 0 0
9| 3 22 51 40 7 0 0 0 0 0
10| 3 25 70 75 25 1 0 0 0 0
11| 3 28 92 126 65 8 0 0 0 0
12| 3 31 117 196 140 33 1 0 0 0
13| 3 34 145 288 266 98 9 0 0 0
14| 3 37 176 405 462 238 42 1 0 0
15| 3 40 210 550 750 504 140 10 0 0
16| 3 43 247 726 1155 966 378 52 1 0
17| 3 46 287 936 1705 1716 882 192 11 0
18| 3 49 330 1183 2431 2871 1848 570 63 1
19| 3 52 376 1470 3367 4576 3564 1452 255 12
Table 3_7_7. All of the diagonals from the (3,1) Pascal Triangle. Source 3,1,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------+-+-----------------------------------------
1 |0| 1 3 1 4 1 5 1 6 1 7
3 |1| 1 3 4 7 11 18 29 47 76 123
3 1 |2| 1 3 7 10 27 37 101 138 377 515
3 4 |3| 1 3 10 13 49 62 235 297 1126 1423
3 7 1 |4| 1 3 13 16 77 93 449 542 2617 3159
3 10 5 |5| 1 3 16 19 111 130 761 891 5216 6107
3 13 12 1 |6| 1 3 19 22 151 173 1189 1362 9361 10723
3 16 22 6 |7| 1 3 22 25 197 222 1751 1973 15562 17535
3 19 35 18 1 |8| 1 3 25 28 249 277 2465 2742 24401 27143
3 22 51 40 7 |9| 1 3 28 31 307 338 3349 3687 36532 40219
A0: p[0](n) = 1, p[1](n) = 3, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 1, a(i,1) = 3, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
3 1 3
+ 3 4 + 3 1
--------- -----
3 7 1 3 4
+ 3 10 5 + 3 7 1
------------- ---------
3 13 12 1 3 10 5
+ 3 16 22 6 + 3 13 12 1
----------------- -------------
3 19 35 18 1 3 16 22 6
Table 10_27_29. The even diagonals from the (3,1) Pascal Triangle. Source 3,1,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------------------------------+-+------------------------------------------------------------------
1 |0| 1 1 1 1 1 1 1 1 1 1
3 1 |1| 1 4 11 29 76 199 521 1364 3571 9349
3 7 1 |2| 1 7 27 101 377 1407 5251 19597 73137 272951
3 13 12 1 |3| 1 10 49 235 1126 5395 25849 123850 593401 2843155
3 19 35 18 1 |4| 1 13 77 449 2617 15253 88901 518153 3020017 17601949
3 25 70 75 25 1 |5| 1 16 111 761 5216 35751 245041 1679536 11511711 78902441
3 31 117 196 140 33 1 |6| 1 19 151 1189 9361 73699 580231 4568149 35964961 283151539
3 37 176 405 462 238 42 1 |7| 1 22 197 1751 15562 138307 1229201 10924502 97091317 862897351
3 43 247 726 1155 966 378 52 1 |8| 1 25 249 2465 24401 241545 2391049 23668945 234298401 2319315065
3 49 330 1183 2431 2871 1848 570 63 1 |9| 1 28 307 3349 36532 398503 4347001 47418508 517256587 5642403949
A0: p[0](n) = 1; p[1](n) = 3*n+1, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = 3*i+1, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
3 7 1
+ 6 14 2
-------------
3 13 15 2
- 3 1
-------------
3 13 12 1
+ 6 26 24 2
-----------------
3 19 38 25 2
- 3 7 1
-----------------
3 19 35 18 1
Table 13_37_47. The odd diagonals from the (3,1) Pascal Triangle. Source 3,1,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------------------------+-+------------------------------------------------------------------
3 |0| 3 4 5 6 7 8 9 10 11 12
3 4 |1| 3 7 18 47 123 322 843 2207 5778 15127
3 10 5 |2| 3 10 37 138 515 1922 7173 26770 99907 372858
3 16 22 6 |3| 3 13 62 297 1423 6818 32667 156517 749918 3593073
3 22 51 40 7 |4| 3 16 93 542 3159 18412 107313 625466 3645483 21247432
3 28 92 126 65 8 |5| 3 19 130 891 6107 41858 286899 1966435 13478146 92380587
3 34 145 288 266 98 9 |6| 3 22 173 1362 10723 84422 664653 5232802 41197763 324349302
3 40 210 550 750 504 140 10 |7| 3 25 222 1973 17535 155842 1385043 12309545 109400862 972298213
3 46 287 936 1705 1716 882 192 11 |8| 3 28 277 2742 27143 268688 2659737 26328682 260627083 2579942148
3 52 376 1470 3367 4576 3564 1452 255 12 |9| 3 31 338 3687 40219 438722 4785723 52204231 569460818 6211864767
A0: p[0](n) = 3, p[1](n) = 3*n+4, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 3, a(i,1) = 3*i+4, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
3 10 5
+ 6 20 10
-------------
3 16 25 10
- 3 4
-------------
3 16 22 6
+ 6 32 44 12
-----------------
3 22 54 50 12
- 3 10 5
-----------------
3 22 51 40 7
============================================================================================================
= =
= Exit build3(3,1) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(1,4) =
= =
============================================================================================================
The (1, 4) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 1 4 0 0 0 0 0 0 0 0
2| 1 5 4 0 0 0 0 0 0 0
3| 1 6 9 4 0 0 0 0 0 0
4| 1 7 15 13 4 0 0 0 0 0
5| 1 8 22 28 17 4 0 0 0 0
6| 1 9 30 50 45 21 4 0 0 0
7| 1 10 39 80 95 66 25 4 0 0
8| 1 11 49 119 175 161 91 29 4 0
9| 1 12 60 168 294 336 252 120 33 4
10| 1 13 72 228 462 630 588 372 153 37
11| 1 14 85 300 690 1092 1218 960 525 190
12| 1 15 99 385 990 1782 2310 2178 1485 715
13| 1 16 114 484 1375 2772 4092 4488 3663 2200
14| 1 17 130 598 1859 4147 6864 8580 8151 5863
15| 1 18 147 728 2457 6006 11011 15444 16731 14014
16| 1 19 165 875 3185 8463 17017 26455 32175 30745
17| 1 20 184 1040 4060 11648 25480 43472 58630 62920
18| 1 21 204 1224 5100 15708 37128 68952 102102 121550
19| 1 22 225 1428 6324 20808 52836 106080 171054 223652
All of the diagonals from the (1, 4) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------
0| 4 0 0 0 0 0 0 0 0 0
1| 1 0 0 0 0 0 0 0 0 0
2| 1 4 0 0 0 0 0 0 0 0
3| 1 5 0 0 0 0 0 0 0 0
4| 1 6 4 0 0 0 0 0 0 0
5| 1 7 9 0 0 0 0 0 0 0
6| 1 8 15 4 0 0 0 0 0 0
7| 1 9 22 13 0 0 0 0 0 0
8| 1 10 30 28 4 0 0 0 0 0
9| 1 11 39 50 17 0 0 0 0 0
10| 1 12 49 80 45 4 0 0 0 0
11| 1 13 60 119 95 21 0 0 0 0
12| 1 14 72 168 175 66 4 0 0 0
13| 1 15 85 228 294 161 25 0 0 0
14| 1 16 99 300 462 336 91 4 0 0
15| 1 17 114 385 690 630 252 29 0 0
16| 1 18 130 484 990 1092 588 120 4 0
17| 1 19 147 598 1375 1782 1218 372 33 0
18| 1 20 165 728 1859 2772 2310 960 153 4
19| 1 21 184 875 2457 4147 4092 2178 525 37
Table 1_6_6. All of the diagonals from the (1,4) Pascal Triangle. Source 1,4,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------+-+-----------------------------------------
4 |0| 4 1 4 5 4 9 4 13 4 17
1 |1| 4 1 5 6 11 17 28 45 73 118
1 4 |2| 4 1 6 7 20 27 74 101 276 377
1 5 |3| 4 1 7 8 31 39 148 187 709 896
1 6 4 |4| 4 1 8 9 44 53 256 309 1492 1801
1 7 9 |5| 4 1 9 10 59 69 404 473 2769 3242
1 8 15 4 |6| 4 1 10 11 76 87 598 685 4708 5393
1 9 22 13 |7| 4 1 11 12 95 107 844 951 7501 8452
1 10 30 28 4 |8| 4 1 12 13 116 129 1148 1277 11364 12641
1 11 39 50 17 |9| 4 1 13 14 139 153 1516 1669 16537 18206
A0: p[0](n) = 4, p[1](n) = 1, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 4, a(i,1) = 1, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
1 4 1
+ 1 5 + 1 4
--------- -----
1 6 4 1 5
+ 1 7 9 + 1 6 4
------------- ---------
1 8 15 4 1 7 9
+ 1 9 22 13 + 1 8 15 4
----------------- -------------
1 10 30 28 4 1 9 22 13
Table 7_20_28. The even diagonals from the (1,4) Pascal Triangle. Source 1,4,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------------------------------+-+------------------------------------------------------------------
4 |0| 4 4 4 4 4 4 4 4 4 4
1 4 |1| 4 5 11 28 73 191 500 1309 3427 8972
1 6 4 |2| 4 6 20 74 276 1030 3844 14346 53540 199814
1 8 15 4 |3| 4 7 31 148 709 3397 16276 77983 373639 1790212
1 10 30 28 4 |4| 4 8 44 256 1492 8696 50684 295408 1721764 10035176
1 12 49 80 45 4 |5| 4 9 59 404 2769 18979 130084 891609 6111179 41886644
1 14 72 168 175 66 4 |6| 4 10 76 598 4708 37066 291820 2297494 18088132 142407562
1 16 99 300 462 336 91 4 |7| 4 11 95 844 7501 66665 592484 5265691 46798735 415922924
1 18 130 484 990 1092 588 120 4 |8| 4 12 116 1148 11364 112492 1113556 11023068 109117124 1080148172
1 20 165 728 1859 2772 2310 960 153 4 |9| 4 13 139 1516 16537 180391 1967764 21465013 234147379 2554156156
A0: p[0](n) = 4; p[1](n) = n+4, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 4, a(i,1) = i+4, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 6 4
+ 2 12 8
-------------
1 8 16 8
- 1 4
-------------
1 8 15 4
+ 2 16 30 8
-----------------
1 10 31 34 8
- 1 6 4
-----------------
1 10 30 28 4
Table 8_27_45. The odd diagonals from the (1,4) Pascal Triangle. Source 1,4,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-----------------------------------------+-+------------------------------------------------------------------
1 |0| 1 5 9 13 17 21 25 29 33 37
1 5 |1| 1 6 17 45 118 309 809 2118 5545 14517
1 7 9 |2| 1 7 27 101 377 1407 5251 19597 73137 272951
1 9 22 13 |3| 1 8 39 187 896 4293 20569 98552 472191 2262403
1 11 39 50 17 |4| 1 9 53 309 1801 10497 61181 356589 2078353 12113529
1 13 60 119 95 21 |5| 1 10 69 473 3242 22221 152305 1043914 7155093 49041737
1 15 85 228 294 161 25 |6| 1 11 87 685 5393 42459 334279 2631773 20719905 163127467
1 17 114 385 690 630 252 29 |7| 1 12 107 951 8452 75117 667601 5933292 52732027 468654951
1 19 147 598 1375 1782 1218 372 33 |8| 1 13 129 1277 12641 125133 1238689 12261757 121378881 1201527053
1 21 184 875 2457 4147 4092 2178 525 37 |9| 1 14 153 1669 18206 198597 2166361 23631374 257778753 2811934909
A0: p[0](n) = 1, p[1](n) = n+5, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = i+5, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
1 7 9
+ 2 14 18
-------------
1 9 23 18
- 1 5
-------------
1 9 22 13
+ 2 18 44 26
-----------------
1 11 40 57 26
- 1 7 9
-----------------
1 11 39 50 17
============================================================================================================
= =
= Exit build3(1,4) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(2,3) =
= =
============================================================================================================
The (2, 3) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+----------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 2 3 0 0 0 0 0 0 0 0
2| 2 5 3 0 0 0 0 0 0 0
3| 2 7 8 3 0 0 0 0 0 0
4| 2 9 15 11 3 0 0 0 0 0
5| 2 11 24 26 14 3 0 0 0 0
6| 2 13 35 50 40 17 3 0 0 0
7| 2 15 48 85 90 57 20 3 0 0
8| 2 17 63 133 175 147 77 23 3 0
9| 2 19 80 196 308 322 224 100 26 3
10| 2 21 99 276 504 630 546 324 126 29
11| 2 23 120 375 780 1134 1176 870 450 155
12| 2 25 143 495 1155 1914 2310 2046 1320 605
13| 2 27 168 638 1650 3069 4224 4356 3366 1925
14| 2 29 195 806 2288 4719 7293 8580 7722 5291
15| 2 31 224 1001 3094 7007 12012 15873 16302 13013
16| 2 33 255 1225 4095 10101 19019 27885 32175 29315
17| 2 35 288 1480 5320 14196 29120 46904 60060 61490
18| 2 37 323 1768 6800 19516 43316 76024 106964 121550
19| 2 39 360 2091 8568 26316 62832 119340 182988 228514
All of the diagonals from the (2, 3) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------
0| 3 0 0 0 0 0 0 0 0 0
1| 2 0 0 0 0 0 0 0 0 0
2| 2 3 0 0 0 0 0 0 0 0
3| 2 5 0 0 0 0 0 0 0 0
4| 2 7 3 0 0 0 0 0 0 0
5| 2 9 8 0 0 0 0 0 0 0
6| 2 11 15 3 0 0 0 0 0 0
7| 2 13 24 11 0 0 0 0 0 0
8| 2 15 35 26 3 0 0 0 0 0
9| 2 17 48 50 14 0 0 0 0 0
10| 2 19 63 85 40 3 0 0 0 0
11| 2 21 80 133 90 17 0 0 0 0
12| 2 23 99 196 175 57 3 0 0 0
13| 2 25 120 276 308 147 20 0 0 0
14| 2 27 143 375 504 322 77 3 0 0
15| 2 29 168 495 780 630 224 23 0 0
16| 2 31 195 638 1155 1134 546 100 3 0
17| 2 33 224 806 1650 1914 1176 324 26 0
18| 2 35 255 1001 2288 3069 2310 870 126 3
19| 2 37 288 1225 3094 4719 4224 2046 450 29
Table 2_7_7. All of the diagonals from the (2,3) Pascal Triangle. Source 2,3,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------+-+-----------------------------------------
3 |0| 3 2 3 5 3 8 3 11 3 14
2 |1| 3 2 5 7 12 19 31 50 81 131
2 3 |2| 3 2 7 9 25 34 93 127 347 474
2 5 |3| 3 2 9 11 42 53 201 254 963 1217
2 7 3 |4| 3 2 11 13 63 76 367 443 2139 2582
2 9 8 |5| 3 2 13 15 88 103 603 706 4133 4839
2 11 15 3 |6| 3 2 15 17 117 134 921 1055 7251 8306
2 13 24 11 |7| 3 2 17 19 150 169 1333 1502 11847 13349
2 15 35 26 3 |8| 3 2 19 21 187 208 1851 2059 18323 20382
2 17 48 50 14 |9| 3 2 21 23 228 251 2487 2738 27129 29867
A0: p[0](n) = 3, p[1](n) = 2, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 3, a(i,1) = 2, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
2 3 2
+ 2 5 + 2 3
--------- -----
2 7 3 2 5
+ 2 9 8 + 2 7 3
------------- ---------
2 11 15 3 2 9 8
+ 2 13 24 11 + 2 11 15 3
----------------- -------------
2 15 35 26 3 2 13 24 11
Table 9_25_31. The even diagonals from the (2,3) Pascal Triangle. Source 2,3,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
----------------------------------------+-+------------------------------------------------------------------
3 |0| 3 3 3 3 3 3 3 3 3 3
2 3 |1| 3 5 12 31 81 212 555 1453 3804 9959
2 7 3 |2| 3 7 25 93 347 1295 4833 18037 67315 251223
2 11 15 3 |3| 3 9 42 201 963 4614 22107 105921 507498 2431569
2 15 35 26 3 |4| 3 11 63 367 2139 12467 72663 423511 2468403 14386907
2 19 63 85 40 3 |5| 3 13 88 603 4133 28328 194163 1330813 9121528 62519883
2 23 99 196 175 57 3 |6| 3 15 117 921 7251 57087 449445 3538473 27858339 219328239
2 27 143 375 504 322 77 3 |7| 3 17 150 1333 11847 105290 935763 8316577 73913430 656904293
2 31 195 638 1155 1134 546 100 3 |8| 3 19 187 1851 18323 181379 1795467 17773291 175937443 1741601139
2 35 255 1001 2288 3069 2310 870 126 3 |9| 3 21 228 2487 27129 295932 3228123 35213421 384119508 4190101167
A0: p[0](n) = 3; p[1](n) = 2*n+3, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 3, a(i,1) = 2*i+3, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
2 7 3
+ 4 14 6
-------------
2 11 17 6
- 2 3
-------------
2 11 15 3
+ 4 22 30 6
-----------------
2 15 37 33 6
- 2 7 3
-----------------
2 15 35 26 3
Table 11_34_50. The odd diagonals from the (2,3) Pascal Triangle. Source 2,3,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------------------------+-+------------------------------------------------------------------
2 |0| 2 5 8 11 14 17 20 23 26 29
2 5 |1| 2 7 19 50 131 343 898 2351 6155 16114
2 9 8 |2| 2 9 34 127 474 1769 6602 24639 91954 343177
2 13 24 11 |3| 2 11 53 254 1217 5831 27938 133859 641357 3072926
2 17 48 50 14 |4| 2 13 76 443 2582 15049 87712 511223 2979626 17366533
2 21 80 133 90 17 |5| 2 15 103 706 4839 33167 227330 1558143 10679671 73199554
2 25 120 276 308 147 20 |6| 2 17 134 1055 8306 65393 514838 4053311 31911650 251239889
2 29 168 495 780 630 224 23 |7| 2 19 169 1502 13349 118639 1054402 9370979 83284409 740188702
2 33 224 806 1650 1914 1176 324 26 |8| 2 21 208 2059 20382 201761 1997228 19770519 195707962 1937309101
2 37 288 1225 3094 4719 4224 2046 450 29 |9| 2 23 251 2738 29867 325799 3553922 38767343 422886851 4612988018
A0: p[0](n) = 2, p[1](n) = 2*n+5, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 2, a(i,1) = 2*i+5, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
2 9 8
+ 4 18 16
-------------
2 13 26 16
- 2 5
-------------
2 13 24 11
+ 4 26 48 22
-----------------
2 17 50 59 22
- 2 9 8
-----------------
2 17 48 50 14
============================================================================================================
= =
= Exit build3(2,3) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(3,2) =
= =
============================================================================================================
The (3, 2) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 3 2 0 0 0 0 0 0 0 0
2| 3 5 2 0 0 0 0 0 0 0
3| 3 8 7 2 0 0 0 0 0 0
4| 3 11 15 9 2 0 0 0 0 0
5| 3 14 26 24 11 2 0 0 0 0
6| 3 17 40 50 35 13 2 0 0 0
7| 3 20 57 90 85 48 15 2 0 0
8| 3 23 77 147 175 133 63 17 2 0
9| 3 26 100 224 322 308 196 80 19 2
10| 3 29 126 324 546 630 504 276 99 21
11| 3 32 155 450 870 1176 1134 780 375 120
12| 3 35 187 605 1320 2046 2310 1914 1155 495
13| 3 38 222 792 1925 3366 4356 4224 3069 1650
14| 3 41 260 1014 2717 5291 7722 8580 7293 4719
15| 3 44 301 1274 3731 8008 13013 16302 15873 12012
16| 3 47 345 1575 5005 11739 21021 29315 32175 27885
17| 3 50 392 1920 6580 16744 32760 50336 61490 60060
18| 3 53 442 2312 8500 23324 49504 83096 111826 121550
19| 3 56 495 2754 10812 31824 72828 132600 194922 233376
All of the diagonals from the (3, 2) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------
0| 2 0 0 0 0 0 0 0 0 0
1| 3 0 0 0 0 0 0 0 0 0
2| 3 2 0 0 0 0 0 0 0 0
3| 3 5 0 0 0 0 0 0 0 0
4| 3 8 2 0 0 0 0 0 0 0
5| 3 11 7 0 0 0 0 0 0 0
6| 3 14 15 2 0 0 0 0 0 0
7| 3 17 26 9 0 0 0 0 0 0
8| 3 20 40 24 2 0 0 0 0 0
9| 3 23 57 50 11 0 0 0 0 0
10| 3 26 77 90 35 2 0 0 0 0
11| 3 29 100 147 85 13 0 0 0 0
12| 3 32 126 224 175 48 2 0 0 0
13| 3 35 155 324 322 133 15 0 0 0
14| 3 38 187 450 546 308 63 2 0 0
15| 3 41 222 605 870 630 196 17 0 0
16| 3 44 260 792 1320 1176 504 80 2 0
17| 3 47 301 1014 1925 2046 1134 276 19 0
18| 3 50 345 1274 2717 3366 2310 780 99 2
19| 3 53 392 1575 3731 5291 4356 1914 375 21
Table 3_8_8. All of the diagonals from the (3,2) Pascal Triangle. Source 3,2,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------+-+-----------------------------------------
2 |0| 2 3 2 5 2 7 2 9 2 11
3 |1| 2 3 5 8 13 21 34 55 89 144
3 2 |2| 2 3 8 11 30 41 112 153 418 571
3 5 |3| 2 3 11 14 53 67 254 321 1217 1538
3 8 2 |4| 2 3 14 17 82 99 478 577 2786 3363
3 11 7 |5| 2 3 17 20 117 137 802 939 5497 6436
3 14 15 2 |6| 2 3 20 23 158 181 1244 1425 9794 11219
3 17 26 9 |7| 2 3 23 26 205 231 1822 2053 16193 18246
3 20 40 24 2 |8| 2 3 26 29 258 287 2554 2841 25282 28123
3 23 57 50 11 |9| 2 3 29 32 317 349 3458 3807 37721 41528
A0: p[0](n) = 2, p[1](n) = 3, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 2, a(i,1) = 3, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
3 2 3
+ 3 5 + 3 2
--------- -----
3 8 2 3 5
+ 3 11 7 + 3 8 2
------------- ---------
3 14 15 2 3 11 7
+ 3 17 26 9 + 3 14 15 2
----------------- -------------
3 20 40 24 2 3 17 26 9
Table 11_30_34. The even diagonals from the (3,2) Pascal Triangle. Source 3,2,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------------------------------+-+------------------------------------------------------------------
2 |0| 2 2 2 2 2 2 2 2 2 2
3 2 |1| 2 5 13 34 89 233 610 1597 4181 10946
3 8 2 |2| 2 8 30 112 418 1560 5822 21728 81090 302632
3 14 15 2 |3| 2 11 53 254 1217 5831 27938 133859 641357 3072926
3 20 40 24 2 |4| 2 14 82 478 2786 16238 94642 551614 3215042 18738638
3 26 77 90 35 2 |5| 2 17 117 802 5497 37677 258242 1770017 12131877 83153122
3 32 126 224 175 48 2 |6| 2 20 158 1244 9794 77108 607070 4779452 37628546 296248916
3 38 187 450 546 308 63 2 |7| 2 23 205 1822 16193 143915 1279042 11367463 101028125 897885662
3 44 260 792 1320 1176 504 80 2 |8| 2 26 258 2554 25282 250266 2477378 24523514 242757762 2403054106
3 50 345 1274 2717 3366 2310 780 99 2 |9| 2 29 317 3458 37721 411473 4488482 48961829 534091637 5826046178
A0: p[0](n) = 2; p[1](n) = 3*n+2, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 2, a(i,1) = 3*i+2, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
3 8 2
+ 6 16 4
-------------
3 14 18 4
- 3 2
-------------
3 14 15 2
+ 6 28 30 4
-----------------
3 20 43 32 4
- 3 8 2
------------------
3 20 40 24 2
Table 14_41_55. The odd diagonals from the (3,2) Pascal Triangle. Source 3,2,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------------------------+-+------------------------------------------------------------------
3 |0| 3 5 7 9 11 13 15 17 19 21
3 5 |1| 3 8 21 55 144 377 987 2584 6765 17711
3 11 7 |2| 3 11 41 153 571 2131 7953 29681 110771 413403
3 17 26 9 |3| 3 14 67 321 1538 7369 35307 169166 810523 3883449
3 23 57 50 11 |4| 3 17 99 577 3363 19601 114243 665857 3880899 22619537
3 29 100 147 85 13 |5| 3 20 137 939 6436 44113 302355 2072372 14204249 97357371
3 35 155 324 322 133 15 |6| 3 23 181 1425 11219 88327 695397 5474849 43103395 339352311
3 41 222 605 870 630 196 17 |7| 3 26 231 2053 18246 162161 1441203 12808666 113836791 1011722453
3 47 301 1014 1925 2046 1134 276 19 |8| 3 29 287 2841 28123 278389 2755767 27279281 270037043 2673091149
3 53 392 1575 3731 5291 4356 1914 375 21 |9| 3 32 349 3807 41528 453001 4941483 53903312 587994949 6414041127
A0: p[0](n) = 3, p[1](n) = 3*n+5, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 3, a(i,1) = 3*i+5, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
3 11 7
+ 6 22 14
-------------
3 17 29 14
- 3 5
-------------
3 17 26 9
+ 6 34 52 18
-----------------
3 23 60 61 18
- 3 11 7
-----------------
3 23 57 50 11
============================================================================================================
= =
= Exit build3(3,2) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(4,1) =
= =
============================================================================================================
The (4, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 4 1 0 0 0 0 0 0 0 0
2| 4 5 1 0 0 0 0 0 0 0
3| 4 9 6 1 0 0 0 0 0 0
4| 4 13 15 7 1 0 0 0 0 0
5| 4 17 28 22 8 1 0 0 0 0
6| 4 21 45 50 30 9 1 0 0 0
7| 4 25 66 95 80 39 10 1 0 0
8| 4 29 91 161 175 119 49 11 1 0
9| 4 33 120 252 336 294 168 60 12 1
10| 4 37 153 372 588 630 462 228 72 13
11| 4 41 190 525 960 1218 1092 690 300 85
12| 4 45 231 715 1485 2178 2310 1782 990 385
13| 4 49 276 946 2200 3663 4488 4092 2772 1375
14| 4 53 325 1222 3146 5863 8151 8580 6864 4147
15| 4 57 378 1547 4368 9009 14014 16731 15444 11011
16| 4 61 435 1925 5915 13377 23023 30745 32175 26455
17| 4 65 496 2360 7840 19292 36400 53768 62920 58630
18| 4 69 561 2856 10200 27132 55692 90168 116688 121550
19| 4 73 630 3417 13056 37332 82824 145860 206856 238238
All of the diagonals from the (4, 1) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 4 0 0 0 0 0 0 0 0 0
2| 4 1 0 0 0 0 0 0 0 0
3| 4 5 0 0 0 0 0 0 0 0
4| 4 9 1 0 0 0 0 0 0 0
5| 4 13 6 0 0 0 0 0 0 0
6| 4 17 15 1 0 0 0 0 0 0
7| 4 21 28 7 0 0 0 0 0 0
8| 4 25 45 22 1 0 0 0 0 0
9| 4 29 66 50 8 0 0 0 0 0
10| 4 33 91 95 30 1 0 0 0 0
11| 4 37 120 161 80 9 0 0 0 0
12| 4 41 153 252 175 39 1 0 0 0
13| 4 45 190 372 336 119 10 0 0 0
14| 4 49 231 525 588 294 49 1 0 0
15| 4 53 276 715 960 630 168 11 0 0
16| 4 57 325 946 1485 1218 462 60 1 0
17| 4 61 378 1222 2200 2178 1092 228 12 0
18| 4 65 435 1547 3146 3663 2310 690 72 1
19| 4 69 496 1925 4368 5863 4488 1782 300 13
Table 4_9_9. All of the diagonals from the (4,1) Pascal Triangle. Source 4,1,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------+-+-----------------------------------------
1 |0| 1 4 1 5 1 6 1 7 1 8
4 |1| 1 4 5 9 14 23 37 60 97 157
4 1 |2| 1 4 9 13 35 48 131 179 489 668
4 5 |3| 1 4 13 17 64 81 307 388 1471 1859
4 9 1 |4| 1 4 17 21 101 122 589 711 3433 4144
4 13 6 |5| 1 4 21 25 146 171 1001 1172 6861 8033
4 17 15 1 |6| 1 4 25 29 199 228 1567 1795 12337 14132
4 21 28 7 |7| 1 4 29 33 260 293 2311 2604 20539 23143
4 25 45 22 1 |8| 1 4 33 37 329 366 3257 3623 32241 35864
4 29 66 50 8 |9| 1 4 37 41 406 447 4429 4876 48313 53189
A0: p[0](n) = 1, p[1](n) = 4, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 1, a(i,1) = 4, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
4 1 4
+ 4 5 + 4 1
--------- -----
4 9 1 4 5
+ 4 13 6 + 4 9 1
------------- ---------
4 17 15 1 4 13 6
+ 4 21 28 7 + 4 17 15 1
----------------- -------------
4 25 45 22 1 4 21 28 7
Table 13_35_37. The even diagonals from the (4,1) Pascal Triangle. Source 4,1,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
---------------------------------------+-+------------------------------------------------------------------
1 |0| 1 1 1 1 1 1 1 1 1 1
4 1 |1| 1 5 14 37 97 254 665 1741 4558 11933
4 9 1 |2| 1 9 35 131 489 1825 6811 25419 94865 354041
4 17 15 1 |3| 1 13 64 307 1471 7048 33769 161797 775216 3714283
4 25 45 22 1 |4| 1 17 101 589 3433 20009 116621 679717 3961681 23090369
4 33 91 95 30 1 |5| 1 21 146 1001 6861 47026 322321 2209221 15142226 103786361
4 41 153 252 175 39 1 |6| 1 25 199 1567 12337 97129 764695 6020431 47398753 373169593
4 49 231 525 588 294 49 1 |7| 1 29 260 2311 20539 182540 1622321 14418349 128142820 1138867031
4 57 325 946 1485 1218 462 60 1 |8| 1 33 329 3257 32241 319153 3159289 31273737 309578081 3064507073
4 65 435 1547 3146 3663 2310 690 72 1 |9| 1 37 406 4429 48313 527014 5748841 62710237 684063766 7461991189
A0: p[0](n) = 1; p[1](n) = 4*n+1, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 1, a(i,1) = 4*i+1, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
4 9 1
+ 8 18 2
-------------
4 17 19 2
- 4 1
-------------
4 17 15 1
+ 8 34 30 2
-----------------
4 25 49 31 2
- 4 9 1
-----------------
4 25 45 22 1
Table 17_48_60. The odd diagonals from the (4,1) Pascal Triangle. Source 4,1,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------------------------+-+------------------------------------------------------------------
4 |0| 4 5 6 7 8 9 10 11 12 13
4 5 |1| 4 9 23 60 157 411 1076 2817 7375 19308
4 13 6 |2| 4 13 48 179 668 2493 9304 34723 129588 483629
4 21 28 7 |3| 4 17 81 388 1859 8907 42676 204473 979689 4693972
4 29 66 50 8 |4| 4 21 122 711 4144 24153 140774 820491 4782172 27872541
4 37 120 161 80 9 |5| 4 25 171 1172 8033 55059 377380 2586601 17728827 121515188
4 45 190 372 336 119 10 |6| 4 29 228 1795 14132 111261 875956 6896387 54295140 427464733
4 53 276 715 960 630 168 11 |7| 4 33 293 2604 23143 205683 1828004 16246353 144389173 1283256204
4 61 378 1222 2200 2178 1092 228 12 |8| 4 37 366 3623 35864 355017 3514306 34788043 344366124 3408873197
4 69 496 1925 4368 5863 4488 1782 300 13 |9| 4 41 447 4876 53189 580203 6329044 69039281 753103047 8215094236
A0: p[0](n) = 4, p[1](n) = 4*n+5, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 4, a(i,1) = 4*i+5, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
4 13 6
+ 8 26 12
-------------
4 21 32 12
- 4 5
-------------
4 21 28 7
+ 8 42 56 14
-----------------
4 29 70 63 14
- 4 13 6
-----------------
4 29 66 50 8
============================================================================================================
= =
= Exit build3(4,1) =
= =
============================================================================================================
============================================================================================================
= =
= Enter build3(3,7) =
= =
============================================================================================================
The (3, 7) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+------------------------------------------------------
0| 1 0 0 0 0 0 0 0 0 0
1| 3 7 0 0 0 0 0 0 0 0
2| 3 10 7 0 0 0 0 0 0 0
3| 3 13 17 7 0 0 0 0 0 0
4| 3 16 30 24 7 0 0 0 0 0
5| 3 19 46 54 31 7 0 0 0 0
6| 3 22 65 100 85 38 7 0 0 0
7| 3 25 87 165 185 123 45 7 0 0
8| 3 28 112 252 350 308 168 52 7 0
9| 3 31 140 364 602 658 476 220 59 7
10| 3 34 171 504 966 1260 1134 696 279 66
11| 3 37 205 675 1470 2226 2394 1830 975 345
12| 3 40 242 880 2145 3696 4620 4224 2805 1320
13| 3 43 282 1122 3025 5841 8316 8844 7029 4125
14| 3 46 325 1404 4147 8866 14157 17160 15873 11154
15| 3 49 371 1729 5551 13013 23023 31317 33033 27027
16| 3 52 420 2100 7280 18564 36036 54340 64350 60060
17| 3 55 472 2520 9380 25844 54600 90376 118690 124410
18| 3 58 527 2992 11900 35224 80444 144976 209066 243100
19| 3 61 585 3519 14892 47124 115668 225420 354042 452166
All of the diagonals from the (3, 7) Pascal Triangle.
i\j| 0 1 2 3 4 5 6 7 8 9
---+-----------------------------------------
0| 7 0 0 0 0 0 0 0 0 0
1| 3 0 0 0 0 0 0 0 0 0
2| 3 7 0 0 0 0 0 0 0 0
3| 3 10 0 0 0 0 0 0 0 0
4| 3 13 7 0 0 0 0 0 0 0
5| 3 16 17 0 0 0 0 0 0 0
6| 3 19 30 7 0 0 0 0 0 0
7| 3 22 46 24 0 0 0 0 0 0
8| 3 25 65 54 7 0 0 0 0 0
9| 3 28 87 100 31 0 0 0 0 0
10| 3 31 112 165 85 7 0 0 0 0
11| 3 34 140 252 185 38 0 0 0 0
12| 3 37 171 364 350 123 7 0 0 0
13| 3 40 205 504 602 308 45 0 0 0
14| 3 43 242 675 966 658 168 7 0 0
15| 3 46 282 880 1470 1260 476 52 0 0
16| 3 49 325 1122 2145 2226 1134 220 7 0
17| 3 52 371 1404 3025 3696 2394 696 59 0
18| 3 55 420 1729 4147 5841 4620 1830 279 7
19| 3 58 472 2100 5551 8866 8316 4224 975 66
Table 3_13_13. All of the diagonals from the (3,7) Pascal Triangle. Source 3,7,0,1.
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
----------------+-+-----------------------------------------
7 |0| 7 3 7 10 7 17 7 24 7 31
3 |1| 7 3 10 13 23 36 59 95 154 249
3 7 |2| 7 3 13 16 45 61 167 228 623 851
3 10 |3| 7 3 16 19 73 92 349 441 1672 2113
3 13 7 |4| 7 3 19 22 107 129 623 752 3631 4383
3 16 17 |5| 7 3 22 25 147 172 1007 1179 6902 8081
3 19 30 7 |6| 7 3 25 28 193 221 1519 1740 11959 13699
3 22 46 24 |7| 7 3 28 31 245 276 2177 2453 19348 21801
3 25 65 54 7 |8| 7 3 31 34 303 337 2999 3336 29687 33023
3 28 87 100 31 |9| 7 3 34 37 367 404 4003 4407 43666 48073
A0: p[0](n) = 7, p[1](n) = 3, n>=0; p[2*j](n) = n * p[2*j-1)(n) + p[2*j-2](n), j>=1,n>=0. zz0
p[2*j+1](n) = p[2*j](n) + p[2*j-1](n), j>=1,n>=0.
A1: a(i,0) = 7, a(i,1) = 3, i>=0; a(i,2*j) = i * a(i,2*j-1) + a(i,2*j-2), j>=1,i>=0.
a(i,2*j+1) = a(i,2*j) + a(i,2*j-1), j>=1,i>=0.
3 7 3
+ 3 10 + 3 7
--------- -----
3 13 7 3 10
+ 3 16 17 + 3 13 7
------------- ---------
3 19 30 7 3 16 17
+ 3 22 46 24 + 3 19 30 7
----------------- -------------
3 25 65 54 7 3 22 46 24
Table 16_45_59. The even diagonals from the (3,7) Pascal Triangle. Source 3,7,0,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-----------------------------------------+-+------------------------------------------------------------------
7 |0| 7 7 7 7 7 7 7 7 7 7
3 7 |1| 7 10 23 59 154 403 1055 2762 7231 18931
3 13 7 |2| 7 13 45 167 623 2325 8677 32383 120855 451037
3 19 30 7 |3| 7 16 73 349 1672 8011 38383 183904 881137 4221781
3 25 65 54 7 |4| 7 19 107 623 3631 21163 123347 718919 4190167 24422083
3 31 112 165 85 7 |5| 7 22 147 1007 6902 47307 324247 2222422 15232707 104406527
3 37 171 364 350 123 7 |6| 7 25 193 1519 11959 94153 741265 5835967 45946471 361735801
3 43 242 675 966 658 168 7 |7| 7 28 245 2177 19348 171955 1528247 13582268 120712165 1072827217
3 49 325 1122 2145 2226 1134 220 7 |8| 7 31 303 2999 29687 293871 2909023 28796359 285054567 2821749311
3 55 420 1729 4147 5841 4620 1830 279 7 |9| 7 34 367 4003 43666 476323 5195887 56678434 618266887 6744257323
A0: p[0](n) = 7; p[1](n) = 3*n+7; n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 7, a(i,1) = 3*i+7, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
3 13 7
+ 6 26 14
-------------
3 19 33 14
- 3 7
-------------
3 19 30 7
+ 6 38 60 14
-----------------
3 25 68 67 14
- 3 13 7
-----------------
3 25 65 54 7
Table 19_61_95. The odd diagonals from the (3,7) Pascal Triangle. Source 3,7,1,2.
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------------------------+-+------------------------------------------------------------------
3 |0| 3 10 17 24 31 38 45 52 59 66
3 10 |1| 3 13 36 95 249 652 1707 4469 11700 30631
3 16 17 |2| 3 16 61 228 851 3176 11853 44236 165091 616128
3 22 46 24 |3| 3 19 92 441 2113 10124 48507 232411 1113548 5335329
3 28 87 100 31 |4| 3 22 129 752 4383 25546 148893 867812 5057979 29480062
3 34 140 252 185 38 |5| 3 25 172 1179 8081 55388 379635 2602057 17834764 122241291
3 40 205 504 602 308 45 |6| 3 28 221 1740 13699 107852 849117 6685084 52631555 414367356
3 46 282 880 1470 1260 476 52 |7| 3 31 276 2453 21801 193756 1722003 15304271 136016436 1208843653
3 52 371 1404 3025 3696 2394 696 59 |8| 3 34 337 3336 33023 326894 3235917 32032276 317086843 3138836154
3 58 472 2100 5551 8866 8316 4224 975 66 |9| 3 37 404 4407 48073 524396 5720283 62398717 680665604 7424922927
A0: p[0](n) = 3, p[1](n) = 3*n+10, n>=0; p[j](n) = (n+2) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz1
A1: a(i,0) = 3, a(i,1) = 3*i+10, i>=0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
3 16 17
+ 6 32 34
-------------
3 22 49 34
- 3 10
-------------
3 22 46 24
+ 6 44 92 48
-----------------
3 28 90 116 48
- 3 16 17
-----------------
3 28 87 100 31
============================================================================================================
= =
= Exit build3(3,7) =
= =
============================================================================================================
============================================================================================================
= =
= Here are Wisteria tables which fell out of the process of debugging. =
= =
= It appears that any table generated by an A0 sequence is also =
= generated by an A1 sequence. Many examples to appear shortly. =
= =
============================================================================================================
.
============================================================================================================
= =
= Table_9_33_67. Parms (1 3,-1,2,2 3). =
= =
============================================================================================================
9 8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------------------------------------------+-+------------------------------------------------------------------
2 |0| 2 3 7 18 47 123 322 843 2207 5778
2 3 |1| 2 5 18 67 250 933 3482 12995 48498 180997
2 9 7 |2| 2 7 33 158 757 3627 17378 83263 398937 1911422
2 15 32 18 |3| 2 9 52 303 1766 10293 59992 349659 2037962 11878113
2 21 75 105 47 |4| 2 11 75 514 3523 24147 165506 1134395 7775259 53292418
2 27 136 315 330 123 |5| 2 13 102 803 6322 49773 391862 3085123 24289122 191227853
2 33 215 702 1200 1008 322 |6| 2 15 133 1182 10505 93363 829762 7374495 65540693 582491742
2 39 312 1320 3170 4293 3016 843 |7| 2 17 168 1663 16462 162957 1613108 15968123 158068122 1564713097
2 45 427 2223 6915 13101 14695 8883 2207 |8| 2 19 207 2258 24631 268683 2930882 31971019 348750327 3804282578
2 51 560 3465 13272 32526 50828 48675 25840 5778 |9| 2 21 250 2979 35498 422997 5040466 60062595 715710674 8528465493
A0: p[0](n) = 2, p[1](n) = 2*n+3, n>=0; p[j](n) = (n+3) * p[j-1](n) - p[j-2](n), j>=2,n>=0. zz*
A1: a(i,0) = 2, a(i,1) = 2*i+3, i>=0; a(i,j) = (i+3) * a(i,j-1) - a(i,j-2), j>=2,i>=0.
2 9 7
+ 6 27 21
-------------
2 15 34 21
- 2 3
-------------
2 15 32 18
+ 6 45 96 54
-----------------
2 21 77 114 54
- 2 9 7
-----------------
2 21 75 105 47
With any object the question to be asked is, "Is it interesting?"
For Wisteria tables the question might translate to "Do many of
the lines in the table occur in OEIS?" For this table the answer
is "not many".
Occurrences of the lines of Table_9_33_67 in OEIS:
0| 2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, 39603
A005248 Bisection of Lucas numbers: a(n) = L(2*n) = A000032(2*n).
1| 2 5 18 67 250 933 3482 12995 48498 180997
A144721 a(0) = 2, a(1) = 5, a(n) = 4 * a(n-1) - a(n-2).
2| 2 7 33 158 757 3627 17378 83263 398937 1911422
See even entries of A237255
3| 2 9 52 303 1766 10293 59992 349659 2037962 11878113
See even entries of A156066
4| 2 11 75 514 3523 24147 165506 1134395 7775259 53292418 nada
5| 2 13 102 803 6322 49773 391862 3085123 24289122 191227853
See A077246.
Bisection (even part) of Chebyshev sequence with Diophantine property.
6| 2 15 133 1182 10505 93363 829762 7374495 65540693 582491742 nada
7| 2 17 168 1663 16462 162957 1613108 15968123 158068122 1564713097 nada
8| 2 19 207 2258 24631 268683 2930882 31971019 348750327 3804282578 nada
9| 2 21 250 2979 35498 422997 5040466 60062595 715710674 8528465493 nada
.
============================================================================================================
= =
= Table 2_5_5_7_12. Parms (1 0, 1, 1, 2). =
= =
============================================================================================================
8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
------------------------+-+--------------------------------------------------------
1 |0| 1 2 1 2 1 2 1 2 1 2
2 |1| 1 2 3 5 8 13 21 34 55 89
2 1 |2| 1 2 5 12 29 70 169 408 985 2378
2 1 2 |3| 1 2 7 23 76 251 829 2738 9043 29867
2 1 4 1 |4| 1 2 9 38 161 682 2889 12238 51841 219602
2 1 6 2 2 |5| 1 2 11 57 296 1537 7981 41442 215191 1117397
2 1 8 3 6 1 |6| 1 2 13 80 493 3038 18721 115364 710905 4380794
2 1 10 4 12 3 2 |7| 1 2 15 107 764 5455 38949 278098 1985635 14177543
2 1 12 5 20 6 8 1 |8| 1 2 17 138 1121 9106 73969 600858 4880833 39647522
2 1 14 6 30 10 20 4 2 |9| 1 2 19 173 1576 14357 130789 1191458 10853911 98876657
A0: p[0](n) = 1; p[1](n) = 2, n>=0; p[j](n) = n * p[j-1](n) + p[j-2](n), j>=2, n>=0. zz*
A1: a(i,0) = 1, a(i,1) = 2, i>=0; a(i,j) = i * a(i,j-1) + a(i,j-2), j>=2, i>=0.
2 1 2
+ 2 1
-------------
2 1 4 1
+ 2 1 2
-----------------
2 1 6 2 2
+ 2 1 4 1
---------------------
2 1 8 3 6 1
.
============================================================================================================
= =
= Table 2_4_5. Parms (1,1 0,1,2). =
= =
============================================================================================================
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------+-+-----------------------------------------
1 |0| 1 2 2 2 2 2 2 2 2 2
2 |1| 1 2 3 5 8 13 21 34 55 89
1 2 |2| 1 2 4 8 16 32 64 128 256 512
3 2 |3| 1 2 5 11 26 59 137 314 725 1667
1 5 2 |4| 1 2 6 14 38 94 246 622 1606 4094
4 7 2 |5| 1 2 7 17 52 137 397 1082 3067 8477
1 9 9 2 |6| 1 2 8 20 68 188 596 1724 5300 15644
5 16 11 2 |7| 1 2 9 23 86 247 849 2578 8521 26567
1 14 25 13 2 |8| 1 2 10 26 106 314 1162 3674 12970 42362
6 30 36 15 2 |9| 1 2 11 29 128 389 1541 5042 18911 64289
A0: p[0](n) = 1; p[1](n) = 2, n>=0; p[j](n) = p[j-1](n) + n * p[j-2](n), j>=2,n>=0. zz*
A1: a(i,0) = 1, a(i,1) = 2, i>=0; a(i,j) = a(i,j-1) + i * a(i,j-2), j>=2,i>=0.
1 2 3 2
+ 3 2 + 1 5 2
--------- -------------
1 5 2 4 7 2
+ 4 7 2 + 1 9 9 2
------------- -----------------
1 9 9 2 5 16 11 2
+ 5 16 11 2 + 1 14 25 13 2
----------------- ---------------------
1 14 25 13 2 6 30 36 15 2
.
============================================================================================================
= =
= Table 1_5_4. Parms (1,1 0,2,1). PT (2,1), 0 mod 1, reversed. =
= =
============================================================================================================
4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
-------------+-+-----------------------------------------
2 |0| 2 1 1 1 1 1 1 1 1 1
1 |1| 2 1 3 4 7 11 18 29 47 76
2 1 |2| 2 1 5 7 17 31 65 127 257 511
3 1 |3| 2 1 7 10 31 61 154 337 799 1810
2 4 1 |4| 2 1 9 13 49 101 297 701 1889 4693
5 5 1 |5| 2 1 11 16 71 151 506 1261 3791 10096
2 9 6 1 |6| 2 1 13 19 97 211 793 2059 6817 19171
7 14 7 1 |7| 2 1 15 22 127 281 1170 3137 11327 33286
2 16 20 8 1 |8| 2 1 17 25 161 361 1649 4537 17729 54025
9 30 27 9 1 |9| 2 1 19 28 199 451 2242 6301 26479 83188
A0: p[0](n) = 2; p[1](n) = 1, n>=0; p[j](n) = p[j-1](n) + n * p[j-2](n), j>=2,n>=0. zz*
A1: a(i,0) = 2, a(i,1) = 1, i>=0; a(i,j) = a(i,j-1) + i * a(i,j-2), j>=2,i>=0.
2
1
-----
2 1
3 1
---------
2 4 1
5 5 1
-------------
2 9 6 1
7 14 7 1
-----------------
2 16 20 8 1
Note that the coefficients for Table 1_5_4 and
Table 1_4_4. (above) are in the opposite order.
.
============================================================================================================
= =
= Table 2_3_-1. Parms (1 0,-1,1,2). =
= =
============================================================================================================
8 7 6 5 4 3 2 1 0 |i| 0 1 2 3 4 5 6 7 8 9
--------------------------------+-+------------------------------------------------------
1 |0| 1 2 -1 -2 1 2 -1 -2 1 2
2 |1| 1 2 1 -1 -2 -1 1 2 1 -1
2 -1 |2| 1 2 3 4 5 6 7 8 9 10
2 -1 -2 |3| 1 2 5 13 34 89 233 610 1597 4181
2 -1 -4 1 |4| 1 2 7 26 97 362 1351 5042 18817 70226
2 -1 -6 2 2 |5| 1 2 9 43 206 987 4729 22658 108561 520147
2 -1 -8 3 6 -1 |6| 1 2 11 64 373 2174 12671 73852 430441 2508794
2 -1 -10 4 12 -3 -2 |7| 1 2 13 89 610 4181 28657 196418 1346269 9227465
2 -1 -12 5 20 -6 -8 1 |8| 1 2 15 118 929 7314 57583 453350 3569217 28100386
2 -1 -14 6 30 -10 -20 4 2 |9| 1 2 17 151 1342 11927 106001 942082 8372737 74412551
A0: p[0](n) = 1; p[1](n) = 2, n>=0; p[j](n) = n * p[j-1](n) - p[j-2](n), j>=2, n>=0. zz*
A1: a(i,0) = 1, a(i,1) = 2, i>=0; a(i,j) = i * a(i,j-1) - a(i,j-2), j>=2, i>=0.
2 -1 -2
- 2 -1
-------------
2 -1 -4 1
- 2 -1 -2
-----------------
2 -1 -6 2 2
- 2 -1 -4 1
---------------------
2 -1 -8 3 6 -1