A217637 -id:A217637 - cp's OEIS frontend

A217631 Number of nX2 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX2 array.

0, 2, 6, 16, 38, 98, 244, 614, 1542, 3872, 9726, 24426, 61348, 154078, 386974, 971904, 2440982, 6130642, 15397396, 38671286, 97124758, 243933408, 612650254, 1538699994, 3864517572, 9705918062, 24376870766, 61223660096, 153766108518

Offset: 0

R. H. Hardin Oct 09 2012

nonn

Also, number of maximal independent sets in the 3-dimensional (2, 2, n) grid graph. [Euler et al.] - N. J. A. Sloane, Nov 21 2013

Column 2 of A217637.

			Some solutions for n=3
..0..0....0..0....0..0....1..1....0..0....1..0....1..0....0..1....1..1....0..0
..0..1....0..0....0..1....0..1....1..0....0..0....0..0....0..0....1..1....1..0
..0..0....1..0....1..1....0..0....0..0....0..0....1..0....0..1....1..1....1..1

R. H. Hardin, Table of n, a(n) for n = 0..210
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; http://www.degruyter.com/view/j/dmgt.2013.33.issue-3/dmgt.1707/dmgt.1707.xml

Cf. A217632, A217637.

G.f. = (2*x+4*x^2+4*x^3)/(1-x-3*x^2-2*x^3). [Euler et al.] - N. J. A. Sloane, Nov 21 2013

Empirical: a(n) = a(n-1) + 3*a(n-2) + 2*a(n-3). (Follows from g.f. - N. J. A. Sloane, Nov 21 2013)

A217632 Number of nX3 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX3 array.

Original entry on oeis.org

0, 4, 16, 66, 244, 968, 3726, 14520, 56352, 218978, 850620, 3304624, 12837742, 49872976, 193747784, 752680930, 2924043092, 11359448344, 44129645550, 171436683864, 666004286592, 2587320999714, 10051331417116, 39047827550656

Offset: 0

R. H. Hardin Oct 09 2012

nonn

Also, number of maximal independent sets in the 3-dimensional (2, 3, n) grid graph. [Euler et al.] - N. J. A. Sloane, Nov 21 2013

Column 3 of A217637.

			Some solutions for n=3
..1..0..0....0..0..0....0..0..0....1..0..0....0..0..1....0..0..1....1..1..0
..0..1..0....0..0..0....0..0..1....0..0..0....0..0..1....0..0..1....1..0..0
..0..0..1....0..1..1....0..0..1....1..0..1....0..0..0....0..0..1....0..0..0

R. H. Hardin, Table of n, a(n) for n = 0..184
R. Euler, P. Oleksik, Z. Skupien, Counting Maximal Distance-Independent Sets in Grid Graphs, Discussiones Mathematicae Graph Theory. Volume 33, Issue 3, Pages 531-557, ISSN (Print) 2083-5892, July 2013; http://www.degruyter.com/view/j/dmgt.2013.33.issue-3/dmgt.1707/dmgt.1707.xml

Cf. A217637.

Empirical: a(n) = 2*a(n-1) +9*a(n-2) -2*a(n-3) -17*a(n-4) -4*a(n-5) +8*a(n-6) -3*a(n-7) +a(n-8) -3*a(n-9) -2*a(n-10) +4*a(n-11)

Euler et al. give an explicit g.f. and recurrence, and so (presumably) prove this recurrence is correct. - N. J. A. Sloane, Nov 21 2013

A217633 Number of n X 4 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 n X 4 array.

Original entry on oeis.org

6, 38, 244, 1418, 8706, 52120, 315378, 1900838, 11472148, 69210290, 417586442, 2519466108, 15201104542, 91714930026, 553357475040, 3338651662490, 20143579337906, 121535200857060, 733276159043790, 4424182455856594, 26693068489330768, 161051201588069830, 971693816818476190

Offset: 1

R. H. Hardin, Oct 09 2012

nonn

			Some solutions for n=3:
..0..1..1..0....0..0..0..0....0..1..0..0....0..1..0..1....0..0..0..1
..0..0..0..0....0..0..0..1....0..0..1..0....0..0..0..0....1..0..0..0
..0..1..0..0....0..0..1..1....0..1..0..0....0..0..0..0....0..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..64

Column 4 of A217637.

Empirical: a(n) = 4*a(n-1) +20*a(n-2) -29*a(n-3) -124*a(n-4) +55*a(n-5) +330*a(n-6) -57*a(n-7) -498*a(n-8) -291*a(n-9) +965*a(n-10) +676*a(n-11) -991*a(n-12) -659*a(n-13) +771*a(n-14) +610*a(n-15) -581*a(n-16) -78*a(n-17) -36*a(n-18) +32*a(n-19) -94*a(n-20) +32*a(n-21) +68*a(n-22) -16*a(n-23).

A217634 Number of nX5 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX5 array.

Original entry on oeis.org

10, 98, 968, 8706, 83074, 773348, 7272142, 68138974, 639248556, 5994907930, 56226693158, 527340415924, 4945869727454, 46386705274750, 435055533728012, 4080334184200118, 38268976277521222, 358920216579635816

Offset: 1

R. H. Hardin Oct 09 2012

nonn

Column 5 of A217637

			Some solutions for n=3
..0..0..1..1..0....0..0..1..0..0....0..0..0..0..0....0..0..0..0..0
..0..0..1..0..0....0..1..0..1..0....0..0..0..1..0....0..1..1..0..0
..0..1..1..0..0....1..0..0..0..0....1..1..1..0..0....0..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..40

A217635 Number of nX6 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX6 array.

Original entry on oeis.org

16, 244, 3726, 52120, 773348, 11181454, 163361868, 2378097084, 34661572702, 505010822224, 7358779655656, 107224919634686, 1562394222954108, 22765852658001192, 331724751544038894, 4833609504204441184

Offset: 1

R. H. Hardin Oct 09 2012

nonn

Column 6 of A217637

			Some solutions for n=3
..0..0..0..0..0..1....0..1..1..0..0..0....0..0..0..0..1..1....1..1..0..0..0..1
..0..1..0..0..1..1....0..0..1..0..0..1....0..0..0..0..1..1....1..0..0..0..1..0
..1..0..0..0..0..1....0..1..0..0..0..0....1..0..0..0..1..1....1..1..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..48

A217636 Number of nX7 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX7 array.

Original entry on oeis.org

26, 614, 14520, 315378, 7272142, 163361868, 3709621842, 83923710538, 1901055652804, 43046530809006, 974841850791586, 22075731493018104, 499920666839751010, 11321017480253759030, 256371943510592766924

Offset: 1

R. H. Hardin Oct 09 2012

nonn

Column 7 of A217637

			Some solutions for n=3
..0..0..0..1..0..0..0....0..1..1..1..1..1..1....0..0..0..0..0..0..0
..0..0..1..1..0..0..0....0..0..0..1..0..1..1....1..0..0..0..0..0..1
..0..0..0..0..0..0..1....0..0..0..0..0..0..1....0..0..0..0..0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..34

A217630 Number of n X n arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 n X n array.

Original entry on oeis.org

2, 6, 66, 1418, 83074, 11181454, 3709621842, 2950299178242, 5681245408101350, 26398983976826626170

Offset: 1

R. H. Hardin Oct 09 2012

nonn

Diagonal of A217637

			Some solutions for n=3
..0..1..1....0..1..1....0..0..0....0..0..1....0..0..1....0..1..0....0..0..0
..0..0..0....0..0..1....0..1..0....0..0..0....0..0..1....0..0..0....0..0..0
..0..0..0....0..0..1....0..0..0....0..0..1....0..0..0....0..0..0....1..0..1

cp's OEIS Frontend

A217631 Number of nX2 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX2 array.

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A217632 Number of nX3 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX3 array.

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A217633 Number of n X 4 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 n X 4 array.

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A217634 Number of nX5 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX5 array.

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A217635 Number of nX6 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX6 array.

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A217636 Number of nX7 arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 nX7 array.

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A217630 Number of n X n arrays of the minimum value of corresponding elements and their horizontal and vertical neighbors in a random 0..1 n X n array.

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