A208555 -id:A208555 - cp's OEIS frontend

A208556 Number of 4 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

9, 81, 225, 1089, 3969, 16641, 65025, 263169, 1046529, 4198401, 16769025, 67125249, 268402689, 1073807361, 4294836225, 17180131329, 68718952449, 274878955521, 1099509530625, 4398050705409, 17592177655809, 70368760954881

Offset: 1

R. H. Hardin, Feb 28 2012

nonn

Row 4 of A208555.

It seems that all terms are squares. - Colin Barker, Mar 07 2018

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

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

Cf. A208555.

Empirical: a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3).
Conjectures from Colin Barker, Mar 07 2018: (Start)
G.f.: 9*x*(1 + 6*x - 8*x^2) / ((1 - x)*(1 + 2*x)*(1 - 4*x)).
a(n) = (2^(n+1) + 1)^2 for n even.
a(n) = (2^(n+1) - 1)^2 for n odd.
(End)

A208550 Number of n X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 16, 90, 1089, 10080, 150544, 1913940, 33930625, 544003170, 11177987076, 215177010048, 5033386477441, 112708053008152, 2958949185458176, 75387859827570360, 2195709810260687361, 62623865365172370000

Offset: 1

R. H. Hardin Feb 28 2012

nonn

Diagonal of A208555.

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

R. H. Hardin, Table of n, a(n) for n = 1..38

Cf. A208555.

A208551 Number of n X 4 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

10, 100, 330, 1089, 2508, 5776, 11020, 21025, 35670, 60516, 94710, 148225, 218680, 322624, 454968, 641601, 873090, 1188100, 1570690, 2076481, 2680260, 3459600, 4376580, 5536609, 6884878, 8561476, 10489710, 12852225, 15544560, 18800896

Offset: 1

R. H. Hardin, Feb 28 2012

nonn

Column 4 of A208555.

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

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

Cf. A208555.

Empirical: a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12).

Empirical g.f.: x*(10 + 80*x + 90*x^2 + 129*x^3 + 60*x^4 + 4*x^5 - 24*x^6 + 6*x^7 + 10*x^8 - 4*x^9 - 2*x^10 + x^11) / ((1 - x)^7*(1 + x)^5). Colin Barker, Jul 03 2018

A208552 Number of n X 5 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

16, 256, 1008, 3969, 10080, 25600, 52000, 105625, 187200, 331776, 536256, 866761, 1310848, 1982464, 2851200, 4100625, 5670000, 7840000, 10502800, 14070001, 18364896, 23970816, 30614688, 39100009, 49023520, 61465600, 75852000, 93605625

Offset: 1

R. H. Hardin, Feb 28 2012

nonn

Column 5 of A208555.

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

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

Cf. A208555.

Empirical: a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12).

Empirical g.f.: x*(16 + 224*x + 432*x^2 + 1089*x^3 + 750*x^4 + 604*x^5 + 90*x^6 + 30*x^7 + 10*x^8 - 4*x^9 - 2*x^10 + x^11) / ((1 - x)^7*(1 + x)^5). - Colin Barker, Jul 04 2018

A208553 Number of nX6 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

26, 676, 3354, 16641, 50052, 150544, 351140, 819025, 1634430, 3261636, 5853246, 10504081, 17449544, 28987456, 45403272, 71115489, 106340130, 159012100, 229010210, 329821921, 460490316, 642926736, 874503084, 1189491121, 1582286342

Offset: 1

R. H. Hardin Feb 28 2012

nonn

Column 6 of A208555

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

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

Empirical: a(n) = 2*a(n-1) +6*a(n-2) -14*a(n-3) -14*a(n-4) +42*a(n-5) +14*a(n-6) -70*a(n-7) +70*a(n-9) -14*a(n-10) -42*a(n-11) +14*a(n-12) +14*a(n-13) -6*a(n-14) -2*a(n-15) +a(n-16)

A208554 Number of nX7 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

42, 1764, 10710, 65025, 221340, 753424, 1913940, 4862025, 10332630, 21958596, 41363322, 77915929, 134523480, 232257600, 375406920, 606784689, 931373730, 1429596100, 2104920510, 3099260241, 4409811252, 6274540944, 8675694300

Offset: 1

R. H. Hardin Feb 28 2012

nonn

Column 7 of A208555

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

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

Empirical: a(n) = 2*a(n-1) +6*a(n-2) -14*a(n-3) -14*a(n-4) +42*a(n-5) +14*a(n-6) -70*a(n-7) +70*a(n-9) -14*a(n-10) -42*a(n-11) +14*a(n-12) +14*a(n-13) -6*a(n-14) -2*a(n-15) +a(n-16).

Empirical formula verified (see link). - Robert Israel, Sep 27 2018

A208557 Number of 5 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

12, 144, 420, 2508, 10080, 50052, 221340, 1042416, 4742628, 21989868, 100900800, 465657348, 2142175548, 9872120016, 45450619620, 209366048652, 964140982560, 4440671206788, 20451041895900, 94190120555184, 433792784654052

Offset: 1

R. H. Hardin, Feb 28 2012

nonn

Row 5 of A208555.

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

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

Cf. A208555.

Empirical: a(n) = a(n-1) + 17*a(n-2) + 6*a(n-3) - 36*a(n-4).

Empirical g.f.: 12*x*(1 + 11*x + 6*x^2 - 36*x^3) / ((1 + x - 3*x^2)*(1 - 2*x - 12*x^2)). - Colin Barker, Jul 04 2018

A208558 Number of 6 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

16, 256, 784, 5776, 25600, 150544, 753424, 4129024, 21492496, 115175824, 607129600, 3230330896, 17097131536, 90759997696, 480986635024, 2551443960976, 13527095526400, 71739070491664, 380392441337104, 2017207032035584

Offset: 1

R. H. Hardin, Feb 28 2012

nonn

Row 6 of A208555.

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

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

Cf. A208555.

Empirical: a(n) = 4*a(n-1) +12*a(n-2) -27*a(n-3).

Empirical g.f.: 16*x*(1 + 12*x - 27*x^2) / ((1 + 3*x)*(1 - 7*x + 9*x^2)). - Colin Barker, Jul 04 2018

A208559 Number of 7 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

20, 400, 1260, 11020, 52000, 351140, 1913940, 11836400, 67894220, 407225740, 2378376000, 14112662980, 82952260180, 490344818000, 2888730695340, 17052623706380, 100542287612000, 593228737061860, 3498693917381460

Offset: 1

R. H. Hardin, Feb 28 2012

nonn

Row 7 of A208555.

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

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

Cf. A208555.

Empirical: a(n) = a(n-1) + 31*a(n-2) + 12*a(n-3) - 144*a(n-4).

Empirical g.f.: 20*x*(1 + 19*x + 12*x^2 - 144*x^3) / (1 - x - 31*x^2 - 12*x^3 + 144*x^4). - Colin Barker, Jul 05 2018

cp's OEIS Frontend

A208556 Number of 4 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208550 Number of n X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208551 Number of n X 4 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208552 Number of n X 5 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208553 Number of nX6 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208554 Number of nX7 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208557 Number of 5 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208558 Number of 6 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A208559 Number of 7 X n 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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