A227165 -id:A227165 - cp's OEIS frontend

A227161 Number of n X 2 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 3 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

1, 3, 8, 18, 36, 66, 113, 183, 283, 421, 606, 848, 1158, 1548, 2031, 2621, 3333, 4183, 5188, 6366, 7736, 9318, 11133, 13203, 15551, 18201, 21178, 24508, 28218, 32336, 36891, 41913, 47433, 53483, 60096, 67306, 75148, 83658, 92873, 102831, 113571, 125133

Offset: 0

R. H. Hardin, Jul 03 2013

nonn

Also number of binary words with 3 1's and at most n 0's that do not contain the substring 101. a(2) = 8: 111, 0111, 1110, 00111, 10011, 11001, 11100, 01110. - Alois P. Heinz, Jul 18 2013

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

R. H. Hardin, Table of n, a(n) for n = 0..210

Column 2 of A227165.

First differences give A177787. - Alois P. Heinz, Jul 18 2013

Empirical: a(n) = (1/24)*n^4 + (1/12)*n^3 + (23/24)*n^2 + (11/12)*n + 1.
G.f.: -(1-x+x^2)^2/(x-1)^5. - Alois P. Heinz, Jul 18 2013
Binomial transform of (1 + 2x + 3x^2 + 2x^3 + x^4), i.e., of (1 + x + x^2)^2. - Gary W. Adamson, Jan 23 2017

a(0) = 1 added by Alois P. Heinz, Jul 18 2013

A227162 Number of n X 3 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 4 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

Original entry on oeis.org

4, 18, 62, 193, 558, 1507, 3828, 9149, 20609, 43918, 88960, 172130, 319637, 572050, 990413, 1664308, 2722302, 4345275, 6783191, 10375943, 15578976, 22994469, 33408938, 47838207, 67580783, 94280764, 130001506, 177311376, 239383023

Offset: 1

R. H. Hardin, Jul 03 2013

nonn

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

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

Column 3 of A227165.

Empirical: a(n) = (1/90720)*n^9 + (1/8064)*n^8 + (17/30240)*n^7 + (13/960)*n^6 - (131/4320)*n^5 + (181/384)*n^4 - (146161/90720)*n^3 + (171511/10080)*n^2 - (25129/504)*n + 58 for n>3.
Conjectures from Colin Barker, Sep 07 2018: (Start)
G.f.: x*(4 - 22*x + 62*x^2 - 97*x^3 + 98*x^4 - 56*x^5 + 32*x^6 - 70*x^7 + 123*x^8 - 113*x^9 + 55*x^10 - 13*x^11 + x^12) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>13.
(End)

A227163 Number of nX4 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X5 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

Original entry on oeis.org

5, 36, 193, 944, 4528, 20336, 85018, 330949, 1200425, 4071133, 12971110, 39024425, 111413038, 303209049, 789787118, 1975994532, 4763501990, 11094968344, 25028901331, 54804425836, 116706101674, 242125356042, 490174804573

Offset: 1

R. H. Hardin Jul 03 2013

nonn

Column 4 of A227165

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

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

Empirical: a(n) = (1/1689515283456000)*n^19 + (1/42682491371520)*n^18 + (47/118562476032000)*n^17 + (289/10461394944000)*n^16 + (479/5230697472000)*n^15 + (7897/2092278988800)*n^14 + (1632263/10461394944000)*n^13 - (908693/603542016000)*n^12 + (896531/18289152000)*n^11 - (23306879/29262643200)*n^10 + (26023484833/1609445376000)*n^9 - (186161720861/804722688000)*n^8 + (2050542947543/871782912000)*n^7 - (5863470751207/392302310400)*n^6 + (78462384223117/1307674368000)*n^5 - (49466529613783/217945728000)*n^4 + (52739410297759/30875644800)*n^3 - (18770426920009/1715313600)*n^2 + (74837531735/2116296)*n - 43518 for n>8

A227164 Number of nX5 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X6 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

Original entry on oeis.org

6, 66, 558, 4528, 37012, 283430, 2010569, 13174529, 79606861, 445291143, 2319506105, 11318251449, 52022806997, 226357815317, 936451009248, 3697654793472, 13982230433991, 50782778554993, 177615062917012

Offset: 1

R. H. Hardin Jul 03 2013

nonn

Column 5 of A227165

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

R. H. Hardin, Table of n, a(n) for n = 1..210
R. H. Hardin, Empirical polynomial of degree 39

Empirical polynomial of degree 39 (see link above)

A227160 Number of n X n 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X(n+1) binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

Original entry on oeis.org

2, 8, 62, 944, 37012, 3754497, 1009485843, 732609096798

Offset: 1

R. H. Hardin Jul 03 2013

nonn

Diagonal of A227165

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

cp's OEIS Frontend

A227161 Number of n X 2 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 3 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

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A227162 Number of n X 3 0,1 arrays indicating 2 X 2 subblocks of some larger (n+1) X 4 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

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A227163 Number of nX4 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X5 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

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A227164 Number of nX5 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X6 binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

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A227160 Number of n X n 0,1 arrays indicating 2X2 subblocks of some larger (n+1)X(n+1) binary array having a sum of one or less, with rows and columns of the latter in lexicographically nondecreasing order.

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