A250729 -id:A250729 - cp's OEIS frontend

A250722 Number of (n+1) X (n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

9, 46, 208, 885, 3605, 14437, 57397, 228455, 912438, 3661515, 14755226, 59674645, 241996099, 983339540, 4001469252, 16299591564, 66441841973, 270971703338, 1105502740041, 4511347592849, 18413367334909, 75165710188419

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

Diagonal of A250729.

			Some solutions for n=4
..0..0..0..0..0....0..0..0..0..0....0..1..0..0..1....0..0..0..0..0
..0..0..0..0..0....0..0..0..0..0....1..0..1..1..1....0..0..0..0..0
..0..0..0..1..0....0..0..0..0..0....0..1..1..1..1....0..0..0..0..1
..0..0..1..0..1....0..0..0..0..0....0..1..1..1..1....0..0..0..0..1
..1..1..0..1..1....1..0..1..0..0....0..1..1..1..1....1..0..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..30

Cf. A250729.

A250723 Number of (n+1) X (2+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

22, 46, 85, 144, 230, 350, 513, 728, 1006, 1358, 1797, 2336, 2990, 3774, 4705, 5800, 7078, 8558, 10261, 12208, 14422, 16926, 19745, 22904, 26430, 30350, 34693, 39488, 44766, 50558, 56897, 63816, 71350, 79534, 88405, 98000, 108358, 119518, 131521

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0....0..1..1....0..0..1....0..0..0....1..0..1....0..0..0....0..0..0
..0..0..1....1..1..1....0..1..1....0..0..0....0..1..1....0..0..0....0..0..0
..1..0..1....1..1..1....0..1..1....0..0..0....1..1..1....0..0..0....0..0..0
..0..1..1....1..1..1....0..1..1....0..0..0....1..1..1....0..0..0....0..0..1
..0..1..1....1..1..1....1..1..1....0..1..1....1..1..1....1..0..0....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 2 of A250729.

Empirical: a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
Empirical for n mod 2 = 0: a(n) = (1/24)*n^4 + (1/2)*n^3 + (10/3)*n^2 + 10*n + 8.
Empirical for n mod 2 = 1: a(n) = (1/24)*n^4 + (1/2)*n^3 + (10/3)*n^2 + 10*n + (65/8).
Empirical g.f.: x*(22 - 42*x + 11*x^2 + 34*x^3 - 31*x^4 + 8*x^5) / ((1 - x)^5*(1 + x)). - Colin Barker, Nov 16 2018

A250724 Number of (n+1) X (3+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

50, 110, 208, 365, 600, 942, 1418, 2065, 2918, 4022, 5420, 7165, 9308, 11910, 15030, 18737, 23098, 28190, 34088, 40877, 48640, 57470, 67458, 78705, 91310, 105382, 121028, 138365, 157508, 178582, 201710, 227025, 254658, 284750, 317440, 352877

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0..1....0..0..0..1....0..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..1....0..0..1..1....0..0..0..0....0..1..1..1....0..0..0..1
..0..1..1..1....0..0..1..1....0..0..0..1....0..1..1..1....0..1..1..1
..1..1..1..1....0..0..1..1....0..0..0..1....0..1..1..1....1..1..1..1
..1..1..1..1....1..0..1..1....0..0..0..1....1..1..1..1....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A250729.

Empirical: a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
Empirical for n mod 2 = 0: a(n) = (1/6)*n^4 + (4/3)*n^3 + (91/12)*n^2 + (74/3)*n + 17.
Empirical for n mod 2 = 1: a(n) = (1/6)*n^4 + (4/3)*n^3 + (91/12)*n^2 + (74/3)*n + (65/4).
Empirical g.f.: x*(50 - 90*x + 18*x^2 + 83*x^3 - 70*x^4 + 17*x^5) / ((1 - x)^5*(1 + x)). - Colin Barker, Nov 16 2018

A250725 Number of (n+1) X (4+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

114, 257, 496, 885, 1500, 2434, 3807, 5760, 8465, 12119, 16954, 23231, 31250, 41344, 53889, 69298, 88031, 110589, 137524, 169433, 206968, 250830, 301779, 360628, 428253, 505587, 593630, 693443, 806158, 932972, 1075157, 1234054, 1411083

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0..1..1....0..0..0..0..0....0..0..0..1..1....0..0..0..0..0
..0..0..0..1..1....0..1..0..0..1....0..0..0..1..1....1..0..0..0..0
..0..0..0..1..1....1..0..1..1..1....0..0..0..1..1....0..1..0..0..0
..1..1..1..1..1....0..1..1..1..1....0..0..0..1..1....1..0..1..0..0
..1..1..1..1..1....1..1..1..1..1....0..1..1..1..1....0..1..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 4 of A250729.

Empirical: a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n>8.
Empirical for n mod 2 = 0: a(n) = (1/60)*n^5 + (13/24)*n^4 + (8/3)*n^3 + (407/24)*n^2 + (3859/60)*n + 30 for n>1.
Empirical for n mod 2 = 1: a(n) = (1/60)*n^5 + (13/24)*n^4 + (8/3)*n^3 + (407/24)*n^2 + (3859/60)*n + (61/2) for n>1.
Empirical g.f.: x*(114 - 313*x + 237*x^2 + 148*x^3 - 316*x^4 + 160*x^5 - 25*x^6 - x^7) / ((1 - x)^6*(1 + x)). - Colin Barker, Nov 16 2018

A250726 Number of (n+1) X (5+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

257, 596, 1158, 2092, 3605, 6016, 9728, 15297, 23407, 34943, 50970, 72810, 102025, 140498, 190420, 254375, 335331, 436729, 562478, 717048, 905469, 1133428, 1407272, 1734109, 2121815, 2579139, 3115714, 3742166, 4470129, 5312358, 6282748, 7396451

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0..0..0..0....0..0..0..1..1..1....0..0..0..0..0..0....0..0..0..0..0..1
..0..0..0..0..0..0....0..0..1..1..1..1....0..0..0..0..0..0....0..0..0..0..0..1
..0..0..0..0..0..0....0..1..1..1..1..1....0..0..0..0..0..0....0..0..0..0..1..1
..0..0..0..0..0..1....0..1..1..1..1..1....0..0..0..0..0..0....0..1..0..0..1..1
..0..0..0..0..0..1....1..1..1..1..1..1....1..1..1..0..1..0....1..0..1..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 5 of A250729.

Empirical: a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8) for n>10.
Empirical for n mod 2 = 0: a(n) = (1/360)*n^6 + (1/12)*n^5 + (49/36)*n^4 + (13/3)*n^3 + (15169/360)*n^2 + (1957/12)*n + 43 for n>2.
Empirical for n mod 2 = 1: a(n) = (1/360)*n^6 + (1/12)*n^5 + (49/36)*n^4 + (13/3)*n^3 + (15169/360)*n^2 + (1957/12)*n + 40 for n>2.
Empirical g.f.: x*(257 - 946*x + 1180*x^2 - 110*x^3 - 1079*x^4 + 1060*x^5 - 440*x^6 + 93*x^7 - 12*x^8 + x^9) / ((1 - x)^7*(1 + x)). - Colin Barker, Nov 16 2018

A250727 Number of (n+1) X (6+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

579, 1376, 2699, 4889, 8514, 14437, 23941, 38821, 61554, 95438, 144820, 215289, 313964, 449751, 633699, 879329, 1203066, 1624648, 2167642, 2859941, 3734372, 4829289, 6189281, 7865869, 9918322, 12414466, 15431616, 19057505, 23391340

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0
..1..0..1..0..1..0..1....0..0..0..0..0..0..0....0..0..0..0..0..0..1
..0..1..0..1..0..1..0....0..0..0..0..0..0..0....0..0..0..0..0..1..0
..1..0..1..0..1..0..1....1..0..0..1..1..1..1....0..0..0..0..1..0..1
..0..1..0..1..0..1..1....0..1..1..1..1..1..1....1..0..0..1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 6 of A250729.

Empirical: a(n) = 7*a(n-1) - 20*a(n-2) + 28*a(n-3) - 14*a(n-4) - 14*a(n-5) + 28*a(n-6) - 20*a(n-7) + 7*a(n-8) - a(n-9) for n>14.
Empirical for n mod 2 = 0: a(n) = (1/2520)*n^7 + (11/720)*n^6 + (179/720)*n^5 + (427/144)*n^4 + (4559/720)*n^3 + (41227/360)*n^2 + (81253/210)*n + 25 for n>5.
Empirical for n mod 2 = 1: a(n) = (1/2520)*n^7 + (11/720)*n^6 + (179/720)*n^5 + (427/144)*n^4 + (4559/720)*n^3 + (41227/360)*n^2 + (81253/210)*n + 27 for n>5.
Empirical g.f.: x*(579 - 2677*x + 4647*x^2 - 2696*x^3 - 2151*x^4 + 4417*x^5 - 2892*x^6 + 866*x^7 - 72*x^8 - 19*x^9 + 9*x^10 - 15*x^11 + 10*x^12 - 2*x^13) / ((1 - x)^8*(1 + x)). - Colin Barker, Nov 16 2018

A250728 Number of (n+1)X(7+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

1302, 3173, 6257, 11377, 19887, 34069, 57397, 95231, 155263, 248537, 390263, 601256, 909242, 1350837, 1973466, 2838027, 4021571, 5620807, 7755707, 10574024, 14256002, 19020095, 25128978, 32896671, 42696063, 54967661, 70228855, 89084528

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

Column 7 of A250729

			Some solutions for n=4
..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..1....0..0..0..0..0..0..0..1
..0..0..0..0..0..0..0..1....0..0..0..0..0..0..0..1....0..0..0..0..0..0..1..1
..0..0..0..0..0..1..1..1....0..0..0..0..0..0..0..1....0..0..0..0..0..1..1..1
..0..0..0..0..0..1..1..1....0..0..0..0..1..0..0..1....0..0..0..0..0..1..1..1
..0..0..0..0..1..1..1..1....1..1..0..1..0..1..0..1....1..0..1..1..0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 8*a(n-1) -27*a(n-2) +48*a(n-3) -42*a(n-4) +42*a(n-6) -48*a(n-7) +27*a(n-8) -8*a(n-9) +a(n-10) for n>17
Empirical for n mod 2 = 0: a(n) = (1/20160)*n^8 + (1/420)*n^7 + (71/1440)*n^6 + (71/120)*n^5 + (17767/2880)*n^4 + (1159/120)*n^3 + (533033/1680)*n^2 + (86336/105)*n - 5 for n>7
Empirical for n mod 2 = 1: a(n) = (1/20160)*n^8 + (1/420)*n^7 + (71/1440)*n^6 + (71/120)*n^5 + (17767/2880)*n^4 + (1159/120)*n^3 + (533033/1680)*n^2 + (86336/105)*n - 17 for n>7

A250730 Number of (1+1)X(n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

9, 22, 50, 114, 257, 579, 1302, 2927, 6578, 14782, 33216, 74637, 167709, 376840, 846753, 1902638, 4275190, 9606266, 21585085, 48501247, 108981314, 244878791, 550237650, 1236372778, 2778104416, 6242343961, 14026419561, 31517078668

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

Row 1 of A250729

			Some solutions for n=4
..0..0..0..1..1....1..0..1..0..1....0..1..0..0..0....1..0..1..0..0
..1..1..0..1..1....0..1..0..1..0....1..0..1..0..1....0..1..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 3*a(n-1) -a(n-2) -2*a(n-3) +a(n-4).

Empirical: G.f. x*( -9+5*x+7*x^2-4*x^3 ) / ( (x-1)*(x^3-x^2-2*x+1) ). - R. J. Mathar, Nov 27 2014

A250731 Number of (2+1) X (n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

18, 46, 110, 257, 596, 1376, 3173, 7310, 16838, 38777, 89300, 205640, 473549, 1090478, 2511134, 5782577, 13315988, 30663728, 70611701, 162602894, 374438006, 862246697, 1985560724, 4572300824, 10528983005, 24245885486, 55832834510

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0..0..1....0..0..0..0..1....0..0..0..0..0....1..0..0..0..0
..0..1..0..0..1....0..1..0..0..1....0..0..0..0..0....0..1..0..0..1
..1..0..1..0..1....1..0..1..1..1....1..0..0..1..0....1..0..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 2 of A250729.

Empirical: a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) for n>4.

Empirical g.f.: x*(18 + 10*x - 18*x^2 - x^3) / ((1 - x)*(1 - x - 3*x^2)). - Colin Barker, Nov 16 2018

A250732 Number of (3+1) X (n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

Original entry on oeis.org

33, 85, 208, 496, 1158, 2699, 6257, 14520, 33640, 77999, 180744, 419005, 971109, 2251135, 5217807, 12095192, 28036004, 64988500, 150642356, 349192782, 809429294, 1876272631, 4349215350, 10081551668, 23369149689, 54170034264

Offset: 1

R. H. Hardin, Nov 27 2014

nonn

			Some solutions for n=4:
..0..0..0..0..0....0..0..0..0..1....0..1..0..0..1....0..0..0..0..1
..0..0..0..0..1....0..0..0..1..1....1..0..1..1..1....0..0..1..1..1
..1..0..0..0..1....0..0..0..1..1....0..1..1..1..1....0..1..1..1..1
..0..1..0..1..1....1..1..0..1..1....1..1..1..1..1....1..1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Row 3 of A250729.

Empirical: a(n) = 4*a(n-1) - 2*a(n-2) - 9*a(n-3) + 12*a(n-4) - 2*a(n-5) - 3*a(n-6) + a(n-7) for n>8.

Empirical g.f.: x*(33 - 47*x - 66*x^2 + 131*x^3 - 41*x^4 - 23*x^5 + 14*x^6 - 2*x^7) / ((1 - x)^2*(1 - 2*x - 3*x^2 + 5*x^3 + x^4 - x^5)). - Colin Barker, Nov 16 2018

cp's OEIS Frontend

A250722 Number of (n+1) X (n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250723 Number of (n+1) X (2+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250724 Number of (n+1) X (3+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250725 Number of (n+1) X (4+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250726 Number of (n+1) X (5+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250727 Number of (n+1) X (6+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250728 Number of (n+1)X(7+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250730 Number of (1+1)X(n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250731 Number of (2+1) X (n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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A250732 Number of (3+1) X (n+1) 0..1 arrays with nondecreasing x(i,j)+x(i,j-1) in the i direction and nondecreasing min(x(i,j),x(i-1,j)) in the j direction.

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