A281205 -id:A281205 - cp's OEIS frontend

A281199 Number of n X 2 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

0, 2, 10, 38, 130, 420, 1308, 3970, 11822, 34690, 100610, 289032, 823800, 2332418, 6566290, 18394910, 51310978, 142587180, 394905492, 1090444930, 3002921270, 8249479162, 22612505090, 61857842448, 168903452400, 460409998850

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Column 2 of A281205.

Empirical: a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
Conjectures from Colin Barker, Feb 16 2019: (Start)
G.f.: 2*x^2*(1 - x) / (1 - 3*x + x^2)^2.
a(n) = (2^(-n)*(2*sqrt(5)*((3-sqrt(5))^n - (3+sqrt(5))^n) - 5*(3-sqrt(5))^n*(1+sqrt(5))*n + 5*(-1+sqrt(5))*(3+sqrt(5))^n*n)) / 25.
(End)

A281200 Number of n X 3 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

1, 14, 56, 168, 448, 1120, 2688, 6272, 14336, 32256, 71680, 157696, 344064, 745472, 1605632, 3440640, 7340032, 15597568, 33030144, 69730304, 146800640, 308281344, 645922816, 1350565888, 2818572288, 5872025600, 12213813248, 25367150592

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 12.

Column 3 of A281205.

Empirical: a(n) = 4*a(n-1) - 4*a(n-2) for n>3.
Conjectures from Colin Barker, Feb 16 2019: (Start)
G.f.: x*(1 + 10*x + 4*x^2) / (1 - 2*x)^2.
a(n) = 7*2^(n-1) * (n-1) for n>1.
(End)

A281201 Number of n X 4 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

2, 28, 98, 270, 676, 1588, 3604, 7960, 17254, 36848, 77776, 162610, 337292, 694982, 1423852, 2902806, 5892558, 11916410, 24017514, 48262212, 96719706, 193358890, 385702166, 767826768, 1525708160, 3026506470, 5994196442, 11854696726

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Column 4 of A281205.

Empirical: a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 6*a(n-4) + 2*a(n-5) + 4*a(n-6) - a(n-8).

Empirical g.f.: 2*x*(1 + x)*(1 + 11*x + 7*x^2 - 8*x^3 - 9*x^4 + 2*x^6) / (1 - x - 2*x^2 + x^4)^2. - Colin Barker, Feb 16 2019

A281202 Number of n X 5 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

5, 52, 176, 470, 1141, 2602, 5712, 12208, 25577, 52784, 107636, 217370, 435473, 866550, 1714460, 3375236, 6616061, 12919308, 25142632, 48783294, 94395997, 182209890, 350933080, 674521464, 1294078657, 2478473672, 4739410828

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Column 5 of A281205.

Empirical: a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-4) + 4*a(n-5) - a(n-8).

Empirical g.f.: x*(5 + 32*x - 12*x^2 - 26*x^3 - 25*x^4 + 2*x^5 + 12*x^6 + 4*x^7) / ((1 - x)^2*(1 - x - x^2 - x^3)^2). - Colin Barker, Feb 17 2019

A281203 Number of n X 6 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

10, 94, 310, 804, 1906, 4248, 9118, 19026, 38916, 78356, 155834, 306840, 599204, 1162074, 2240438, 4297644, 8207494, 15613762, 29601530, 55948952, 105457480, 198283598, 371980528, 696408816, 1301351164, 2427600480, 4521378510

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Column 6 of A281205.

Empirical: a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 6*a(n-4) + 4*a(n-6) + 6*a(n-7) + 3*a(n-8) - 2*a(n-9) - 3*a(n-10) - 2*a(n-11) - a(n-12).

Empirical g.f.: 2*x*(1 + x)*(5 + 32*x + 14*x^2 - 43*x^3 - 55*x^4 - 31*x^5 + x^6 + 28*x^7 + 24*x^8 + 10*x^9 + 3*x^10) / (1 - x - 2*x^2 + x^4 + x^5 + x^6)^2. - Colin Barker, Feb 17 2019

A281204 Number of n X 7 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

20, 166, 537, 1358, 3137, 6838, 14375, 29416, 58984, 116466, 227134, 438532, 839659, 1596460, 3017310, 5673426, 10619999, 19801344, 36791933, 68149580, 125882603, 231941798, 426388234, 782226746, 1432314394, 2618121324, 4778002317

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Column 7 of A281205.

Empirical: a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-4) + 4*a(n-5) - 2*a(n-6) + 4*a(n-7) - a(n-8) - 2*a(n-10) - a(n-12).

Empirical g.f.: x*(20 + 86*x - 47*x^2 - 126*x^3 - 107*x^4 - 26*x^5 + 21*x^6 + 88*x^7 + 92*x^8 + 56*x^9 + 23*x^10 + 10*x^11) / ((1 - 2*x + x^2 - x^3)^2*(1 - x^2 - x^3)^2). - Colin Barker, Feb 18 2019

A281206 Number of 2 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 2, 14, 28, 52, 94, 166, 290, 502, 864, 1480, 2526, 4298, 7294, 12350, 20868, 35196, 59262, 99630, 167258, 280422, 469576, 785424, 1312318, 2190482, 3652854, 6086126, 10131820, 16853572, 28013854, 46531510, 77237906, 128126038, 212413104

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Row 2 of A281205.

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

Empirical g.f.: 2*x^2*(1 + 4*x - 6*x^2 - 6*x^3 + 3*x^4 + 2*x^5) / ((1 - x)*(1 - x - x^2)^2). - Colin Barker, Feb 18 2019

A281207 Number of 3 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 10, 56, 98, 176, 310, 537, 922, 1573, 2672, 4524, 7640, 12875, 21658, 36375, 61006, 102184, 170954, 285693, 476954, 795497, 1325596, 2207076, 3671788, 6103951, 10139930, 16833147, 27926522, 46302368, 76725022, 127066209, 210326170

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Row 3 of A281205.

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

Empirical g.f.: x^2*(10 + 26*x - 60*x^2 - 32*x^3 + 38*x^4 + 11*x^5 - 5*x^6) / ((1 - x)*(1 - x - x^2)^2). - Colin Barker, Feb 18 2019

A281208 Number of 4 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 38, 168, 270, 470, 804, 1358, 2284, 3834, 6432, 10786, 18080, 30290, 50712, 84838, 141812, 236846, 395228, 658966, 1097796, 1827410, 3039624, 5052282, 8391768, 13929370, 23106544, 38306878, 63470044, 105104774, 173959572, 287777246

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Row 4 of A281205.

Empirical: a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6) for n>10.

Empirical g.f.: 2*x^2*(19 + 8*x - 125*x^2 + 69*x^3 + 94*x^4 - 55*x^5 - 17*x^6 + 13*x^7 + x^8) / ((1 - x)^2*(1 - x - x^2)^2). - Colin Barker, Feb 18 2019

A281209 Number of 5 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Original entry on oeis.org

0, 130, 448, 676, 1141, 1906, 3137, 5160, 8510, 14084, 23379, 38894, 64795, 108022, 180126, 300318, 500525, 833770, 1388053, 2309332, 3839550, 6379528, 10592935, 17578086, 29151695, 48317346, 80038622, 132514530, 219282785, 362687074

Offset: 1

R. H. Hardin, Jan 17 2017

nonn

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

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

Row 5 of A281205.

Empirical: a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6) for n>11.

Empirical g.f.: x^2*(130 - 72*x - 596*x^2 + 489*x^3 + 422*x^4 - 363*x^5 - 56*x^6 + 114*x^7 + 10*x^8 - 4*x^9) / ((1 - x)^2*(1 - x - x^2)^2). - Colin Barker, Feb 18 2019

cp's OEIS Frontend

A281199 Number of n X 2 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281200 Number of n X 3 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281201 Number of n X 4 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281202 Number of n X 5 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281203 Number of n X 6 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281204 Number of n X 7 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281206 Number of 2 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281207 Number of 3 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281208 Number of 4 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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A281209 Number of 5 X n 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

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