A297299 -id:A297299 - cp's OEIS frontend

A297297 Number of nX6 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

1, 6, 17, 50, 157, 495, 1574, 5031, 16123, 51749, 166241, 534301, 1717724, 5523165, 17760715, 57115518, 183679098, 590706934, 1899713778, 6109510235, 19648337645, 63189642502, 203219959503, 653562362810, 2101879559822, 6759719676624

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Column 6 of A297299.

			Some solutions for n=5
..0..0..1..1..1..1. .0..1..1..0..0..0. .0..1..1..0..0..0. .0..0..0..0..0..0
..0..1..0..0..1..1. .1..1..0..0..0..0. .1..1..0..1..1..1. .0..0..1..1..1..1
..1..1..1..0..1..1. .0..0..0..1..1..1. .0..0..1..0..1..1. .0..1..0..0..1..1
..0..1..0..0..1..0. .0..0..1..0..1..0. .0..1..0..0..1..0. .1..1..1..0..1..0
..1..1..1..1..0..0. .0..1..1..1..0..0. .1..1..1..1..0..0. .0..1..1..1..0..0

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

Cf. A297299.

Empirical: a(n) = 4*a(n-1) -8*a(n-3) -2*a(n-4) +4*a(n-5) +4*a(n-6) +a(n-7) -a(n-8)

A297298 Number of nX7 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 9, 32, 130, 600, 2816, 13504, 65521, 319835, 1566612, 7687400, 37759082, 185561978, 912173296, 4484666199, 22050441495, 108423345160, 533136119034, 2621552944596, 12890861627884, 63387955860276, 311696797461453

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Column 7 of A297299.

			Some solutions for n=5
..0..0..0..0..0..0..0. .0..1..1..0..0..0..0. .0..0..0..1..1..1..0
..0..0..1..1..1..1..0. .1..1..0..0..0..0..0. .0..0..1..0..1..1..1
..0..1..0..0..1..1..1. .0..0..0..0..0..0..0. .0..1..0..0..0..1..1
..1..1..1..0..0..1..0. .0..1..1..0..0..1..1. .1..1..1..1..0..1..0
..0..1..1..1..1..0..0. .1..1..0..0..1..1..0. .0..0..1..1..1..0..0

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

Cf. A297299.

Empirical: a(n) = 7*a(n-1) -6*a(n-2) -26*a(n-3) +18*a(n-4) +37*a(n-5) -25*a(n-7) -9*a(n-8) -6*a(n-9) -2*a(n-10) +6*a(n-11) +5*a(n-12) for n>14

A297295 Number of n X n 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1's.

Original entry on oeis.org

1, 1, 3, 10, 49, 495, 13504, 629563, 40753529, 4162842319, 790491998237, 268041967942170, 155466640607601204, 159960976273554094907, 300474954115694463716874, 1003924486221768324971576564

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Diagonal of A297299.

			Some solutions for n=5
..0..0..1..1..0. .0..0..0..0..0. .0..0..0..1..1. .0..1..1..0..0
..0..1..1..0..0. .0..0..0..0..0. .0..0..1..1..0. .1..1..0..0..0
..0..0..0..0..0. .0..1..1..0..0. .0..0..0..0..0. .0..1..1..0..0
..0..0..1..1..0. .1..1..0..1..1. .0..0..0..0..0. .1..1..0..1..1
..0..1..1..0..0. .0..0..1..1..0. .0..0..0..0..0. .0..0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..21

Cf. A297299.

A297296 Number of n X 5 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 4, 9, 21, 49, 114, 266, 620, 1446, 3372, 7864, 18340, 42772, 99752, 232640, 542560, 1265352, 2951040, 6882384, 16051024, 37434032, 87303264, 203607776, 474851968, 1107444896, 2582771648, 6023513600, 14047976768, 32762547648, 76408478336

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

			Some solutions for n=5:
..0..0..0..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..0..0..1..1
..0..0..1..1..0. .0..0..1..1..1. .0..0..0..0..0. .0..0..1..1..0
..0..1..1..0..0. .0..1..0..1..0. .0..0..0..0..0. .0..0..0..1..1
..0..0..0..0..0. .1..1..1..0..0. .0..0..0..1..1. .0..0..1..1..0
..0..0..0..0..0. .0..0..0..0..0. .0..0..1..1..0. .0..0..0..0..0

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

Column 5 of A297299.

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

Empirical g.f.: x*(1 + x)*(1 + x - 2*x^2 - x^3) / (1 - 2*x - 2*x^2 + 2*x^3 + 2*x^4). - Colin Barker, Feb 26 2019

A297300 Number of 3 X n 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 1, 3, 5, 9, 17, 32, 60, 113, 213, 401, 755, 1422, 2678, 5043, 9497, 17885, 33681, 63428, 119448, 224945, 423617, 797757, 1502339, 2829210, 5327978, 10033667, 18895437, 35583953, 67011825, 126196904, 237654452, 447551697, 842830925

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

			Some solutions for n=5:
..0..1..1..0..0. .0..0..0..0..0. .0..0..0..0..0. .0..1..1..0..0
..1..1..0..1..1. .0..0..1..1..0. .0..0..0..0..0. .1..1..0..0..0
..0..0..1..1..0. .0..1..1..0..0. .0..0..0..0..0. .0..0..0..0..0

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

Row 3 of A297299.

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

Empirical g.f.: x*(1 - x)*(1 + x^2)^2 / (1 - 2*x + x^2 - 2*x^3 + x^4). - Colin Barker, Feb 26 2019

A297301 Number of 4 X n 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 1, 5, 10, 21, 50, 130, 332, 840, 2128, 5408, 13772, 35102, 89465, 227961, 580788, 1479748, 3770356, 9606981, 24478804, 62372090, 158923930, 404938148, 1031783666, 2628989168, 6698674670, 17068245866, 43489945955, 110812526805

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Row 4 of A297299.

			Some solutions for n=5:
  0 0 0 0 0    0 0 0 0 0    0 0 1 1 1    0 1 1 0 0
  0 0 0 0 0    0 0 0 0 0    0 1 0 1 1    1 1 0 0 0
  0 0 1 1 0    0 0 0 1 1    1 1 0 1 0    0 0 0 1 1
  0 1 1 0 0    0 0 1 1 0    1 1 1 0 0    0 0 1 1 0

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical formula

Cf. A297299.

Empirical: a(n) = 4*a(n-1) - 6*a(n-2) + 7*a(n-3) - 4*a(n-4) + 3*a(n-5) + a(n-6) - 5*a(n-7) + 4*a(n-8) - 2*a(n-9) - a(n-10) - 5*a(n-12) - 3*a(n-13) for n > 16.

Formula verified by Robert Israel, May 25 2018: see link.

A297302 Number of 5Xn 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 1, 8, 19, 49, 157, 600, 2161, 7479, 25639, 88307, 306281, 1066691, 3716461, 12931923, 44953528, 156232914, 543075295, 1888104965, 6564738537, 22824408491, 79353796243, 275886391011, 959166284191, 3334724626513, 11593837466750

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Row 5 of A297299.

			Some solutions for n=5
..0..0..0..1..1. .0..0..0..0..0. .0..0..0..0..0. .0..1..1..0..0
..0..0..1..1..0. .0..0..0..1..1. .0..1..1..0..0. .1..1..0..0..0
..0..0..0..0..0. .0..0..1..1..0. .1..1..0..0..0. .0..0..1..1..0
..0..0..0..0..0. .0..0..0..1..1. .0..0..0..0..0. .0..1..1..0..0
..0..0..0..0..0. .0..0..1..1..0. .0..0..0..0..0. .0..0..0..0..0

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

Cf. A297299.

Empirical: a(n) = 7*a(n-1) -21*a(n-2) +42*a(n-3) -56*a(n-4) +67*a(n-5) -40*a(n-6) -48*a(n-7) +185*a(n-8) -212*a(n-9) +185*a(n-10) -144*a(n-11) -37*a(n-12) +221*a(n-13) -238*a(n-14) -59*a(n-15) +a(n-16) +399*a(n-17) -152*a(n-18) -295*a(n-19) +284*a(n-20) +163*a(n-21) -349*a(n-22) -328*a(n-23) +203*a(n-24) +273*a(n-25) +58*a(n-26) -102*a(n-27) -59*a(n-28) +8*a(n-29) +11*a(n-30) +4*a(n-31) -2*a(n-32) for n>37

A297303 Number of 6Xn 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 1, 13, 36, 114, 495, 2816, 14138, 64406, 288079, 1310914, 6040052, 27971466, 129714539, 601185145, 2783395440, 12878290842, 59574077350, 275601620452, 1275129562555, 5900033896489, 27299793712340, 126316193270001

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Row 6 of A297299.

			Some solutions for n=5
..0..1..1..0..0. .0..0..1..1..0. .0..0..1..1..1. .0..0..0..0..0
..1..1..0..0..0. .0..1..1..0..0. .0..1..0..1..1. .0..0..0..0..0
..0..1..1..0..0. .0..0..0..0..0. .1..1..0..1..0. .0..0..0..0..0
..1..1..0..1..1. .0..0..0..0..0. .1..1..1..0..0. .0..0..0..0..0
..0..0..1..1..0. .0..0..0..0..0. .0..0..1..1..0. .0..1..1..0..0
..0..0..0..0..0. .0..0..0..0..0. .0..1..1..0..0. .1..1..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical recurrence of order 68

Cf. A297299.

Empirical recurrence of order 68 (see link above)

A297304 Number of 7Xn 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

Original entry on oeis.org

1, 1, 21, 69, 266, 1574, 13504, 93906, 551477, 3159693, 18792129, 113862870, 692747205, 4226551294, 25828653795, 157805371000, 963306137176, 5876339680839, 35834865220658, 218515617055715, 1332575380148292

Offset: 1

R. H. Hardin, Dec 27 2017

nonn

Row 7 of A297299.

			Some solutions for n=5
..0..1..1..0..0. .0..0..1..1..0. .0..0..0..1..1. .0..0..0..0..0
..1..1..0..0..0. .0..1..1..0..0. .0..0..1..1..0. .0..1..1..0..0
..0..0..0..0..0. .0..0..1..1..1. .0..0..0..0..0. .1..1..0..0..0
..0..0..0..0..0. .0..1..0..1..1. .0..1..1..0..0. .0..1..1..0..0
..0..0..0..0..0. .1..1..0..1..0. .1..1..0..1..1. .1..1..0..0..0
..0..0..0..1..1. .1..1..1..0..0. .0..0..1..1..0. .0..1..1..0..0
..0..0..1..1..0. .0..0..0..0..0. .0..0..0..0..0. .1..1..0..0..0

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

Cf. A297299.

cp's OEIS Frontend

A297297 Number of nX6 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A297298 Number of nX7 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A297295 Number of n X n 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1's.

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A297296 Number of n X 5 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A297300 Number of 3 X n 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A297301 Number of 4 X n 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A297302 Number of 5Xn 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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A297303 Number of 6Xn 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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Formula

A297304 Number of 7Xn 0..1 arrays with every 1 horizontally or antidiagonally adjacent to 2 neighboring 1s.

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