A185825 -id:A185825 - cp's OEIS frontend

A185818 1/5 the number of n X 2 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

1, 9, 76, 656, 5680, 49248, 426928, 3701360, 32089696, 278208816, 2411993584, 20911320416, 181295389360, 1571781109104, 13626909445216, 118141552910384, 1024254735084784, 8880006538838880, 76987211704914352, 667457928119357552

Offset: 1

R. H. Hardin, Feb 05 2011

nonn

Column 2 of A185825.

			Some solutions for 4 X 2 with a(1,1)=0:
..0..2....0..0....0..0....0..0....0..0....0..0....0..3....0..0....0..0....0..0
..0..2....1..1....0..0....0..3....3..2....2..0....0..3....3..4....0..2....0..3
..1..1....1..1....4..4....4..3....3..2....2..0....2..3....3..4....4..2....3..3
..0..0....0..0....3..3....4..3....3..3....1..1....2..2....3..4....4..2....2..2

R. H. Hardin, Table of n, a(n) for n = 1..200
Robert Israel, Maple-assisted proof of formula

Cf. A185825.

f:= gfun:-rectoproc({a(n) = 7*a(n-1) + 15*a(n-2) - 32*a(n-4) - 64*a(n-5), a(1)=1, a(2)=9, a(3)=76, a(4)=656, a(5)=5680},a(n),remember):
map(f, [$1..30]); # Robert Israel, Jul 23 2018

x='x+O('x^99); Vec(x*(1+2*x-2*x^2-11*x^3-20*x^4)/(1-7*x-15*x^2+32*x^4+64*x^5)) \\ Altug Alkan, Jul 23 2018

Empirical: a(n) = 7*a(n-1) + 15*a(n-2) - 32*a(n-4) - 64*a(n-5).
Empirical g.f.: x*(1 + 2*x - 2*x^2 - 11*x^3 - 20*x^4) / (1 - 7*x - 15*x^2 + 32*x^4 + 64*x^5). - Colin Barker, Apr 16 2018
Empirical formulas verified (see link). - Robert Israel, Jul 23 2018

A185819 1/5 the number of nX3 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

1, 76, 1584, 49036, 1266624, 35446376, 956312244, 26231137608, 714107480512, 19507209473948, 532049001583280, 14521612514267224, 396222067735245796, 10812492512879834968, 295042237941175359056, 8051107927651445871340

Offset: 1

R. H. Hardin Feb 05 2011

nonn

Column 3 of A185825

			Some solutions for 4X3 with a(1,1)=0
..0..1..1....0..3..3....0..3..1....0..0..1....0..2..3....0..0..2....0..0..2
..0..1..0....0..4..3....0..3..1....0..1..1....0..2..3....0..2..2....4..4..2
..0..0..0....4..4..2....0..2..2....2..1..3....0..2..4....4..4..4....1..4..4
..1..1..1....4..2..2....1..1..2....2..1..3....0..2..4....1..1..1....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=17*a(n-1)+327*a(n-2)-1020*a(n-3)-6773*a(n-4)+2794*a(n-5)+32525*a(n-6)+131655*a(n-7)-337771*a(n-8)+33788*a(n-9)+517885*a(n-10)-1145646*a(n-11)-5007180*a(n-12)+7715688*a(n-13)+39324336*a(n-14)-82942336*a(n-15)-35554304*a(n-16)+265939200*a(n-17)+170422272*a(n-18)-172261376*a(n-19)-774750208*a(n-20)-958660608*a(n-21)-1396703232*a(n-22)-1811939328*a(n-23)

A185820 1/5 the number of nX4 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

5, 656, 49036, 4443292, 378212944, 32995092992, 2852526229300, 247462638659680, 21440004040979204, 1858466937288261508, 161065823451839652032, 13959918564726559991320, 1209903218593886396425592

Offset: 1

R. H. Hardin Feb 05 2011

nonn

Column 4 of A185825

			Some solutions for 6X4 with a(1,1)=0
..0..0..0..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..1..1..0....0..1..1..0....0..1..1..0
..3..0..0..0....3..0..0..0....1..0..0..0....1..0..0..0....0..0..0..0
..3..1..1..2....3..3..2..2....1..2..2..2....1..1..2..2....2..1..2..2
..4..0..0..2....4..3..1..1....1..0..0..2....3..2..2..1....2..1..4..4
..4..2..2..2....4..3..3..3....1..4..4..2....3..3..1..1....3..3..2..2

R. H. Hardin, Table of n, a(n) for n = 1..106

A185821 1/5 the number of n X 5 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

9, 5680, 1266624, 378212944, 100480040192, 28043363244452, 7662203273902712, 2112817100467905536, 580282378007177354752, 159650092398206756238532, 43890600560593534666234904, 12070254095603726713513018032

Offset: 1

R. H. Hardin, Feb 05 2011

nonn

Column 5 of A185825.

			Some solutions for 4 X 5 with a(1,1)=0
..0..0..0..2..2....0..0..4..4..0....0..0..3..3..2....0..0..3..3..2
..0..1..0..4..4....0..1..3..4..0....0..1..3..3..2....0..1..0..0..2
..0..1..0..3..3....0..1..3..3..4....0..1..0..2..4....0..1..0..2..4
..0..0..0..1..1....0..0..0..0..4....0..0..0..2..4....0..0..0..2..4

Cf. A185825.

A185822 1/5 the number of nX6 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

29, 49248, 35446376, 32995092992, 28043363244452

Offset: 1

R. H. Hardin Feb 05 2011

nonn

Column 6 of A185825

			Some solutions for 4X6 with a(1,1)=0
..0..0..1..1..4..4....0..0..2..2..0..0....0..0..4..4..2..2....0..0..0..3..2..3
..0..1..0..2..2..4....0..1..0..0..0..1....0..1..0..4..1..2....0..1..0..3..2..3
..0..1..0..2..0..2....0..1..0..2..1..1....0..1..0..2..1..2....0..1..0..2..4..4
..0..0..0..2..0..2....0..0..0..2..4..4....0..0..0..2..3..3....0..0..0..2..4..4

A185823 1/5 the number of n X 7 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

65, 426928, 956312244, 2852526229300, 7662203273902712

Offset: 1

R. H. Hardin, Feb 05 2011

nonn

Column 7 of A185825.

			Some solutions for 4 X 7 with a(1,1)=0
..0..0..0..3..3..1..2....0..0..0..2..2..2..0....0..0..0..1..4..4..0
..0..1..0..1..2..1..2....0..1..0..1..1..0..0....0..1..0..1..3..3..0
..0..1..0..1..2..1..2....0..1..0..1..0..1..1....0..1..0..1..0..4..4
..0..0..0..1..0..0..2....0..0..0..1..0..0..0....0..0..0..1..0..3..3

Cf. A185825.

A185824 1/5 the number of nX8 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

Original entry on oeis.org

181, 3701360, 26231137608, 247462638659680, 2112817100467905536

Offset: 1

R. H. Hardin Feb 05 2011

nonn

Column 8 of A185825

			Some solutions for 4X8 with a(1,1)=0
..0..0..0..0..2..2..2..2....0..0..0..0..4..4..3..3....0..0..0..0..3..3..2..2
..0..1..1..0..0..3..3..1....0..1..1..0..0..1..3..3....0..1..1..0..0..3..3..3
..0..0..0..1..1..0..0..1....0..0..0..1..1..1..4..4....0..0..0..1..1..2..2..2
..1..1..0..0..0..0..1..1....1..1..0..0..0..0..0..4....1..1..0..0..0..0..0..2

cp's OEIS Frontend

A185818 1/5 the number of n X 2 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A185819 1/5 the number of nX3 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A185820 1/5 the number of nX4 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A185821 1/5 the number of n X 5 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A185822 1/5 the number of nX6 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A185823 1/5 the number of n X 7 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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A185824 1/5 the number of nX8 0..4 arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.

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