A059679 -id:A059679 - cp's OEIS frontend

A034184 Not necessarily symmetric n X 3 crossword puzzle grids.

1, 15, 111, 649, 3495, 18189, 93231, 474479, 2406621, 12187137, 61668609, 311938233, 1577602849, 7977940187, 40342860995, 204001993697, 1031568839407, 5216271035257, 26376744398811, 133377264694375

Offset: 1

Erich Friedman

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.

Row 3 of A292357.
Column sums of A059679.
Cf. A001333, A034182, A034184, A034185, A034186, A034187.

Appears to obey a 9-term linear recurrence. - Ralf Stephan, May 05 2004

Empirical g.f.: -x*(x^8-x^7-2*x^6-9*x^5+14*x^4-11*x^3-6*x^2+5*x+1) / ((x-1)*(x^2+2*x-1)*(x^6-7*x^5+x^4+6*x^3-11*x^2+7*x-1)). - Colin Barker, Jun 09 2013

A059680 Triangle T(n,k) read by rows giving number of fixed 4 X k polyominoes with n cells (n >= 4, 1<=k<=n-3).

Original entry on oeis.org

1, 0, 12, 0, 18, 50, 0, 8, 154, 120, 0, 1, 212, 584, 230, 0, 0, 158, 1396, 1526, 388, 0, 0, 62, 2038, 5154, 3276, 602, 0, 0, 12, 1952, 11328, 14192, 6194, 880, 0, 0, 1, 1232, 17598, 41196, 32824, 10704, 1230, 0, 0, 0, 488, 19912, 87980, 117616, 67284, 17294, 1660

Offset: 4

N. J. A. Sloane, Feb 05 2001

			Triangle starts:
1;
0, 12;
0, 18,  50;
0,  8, 154,  120;
0,  1, 212,  584,   230;
0,  0, 158, 1396,  1526,   388;
0,  0,  62, 2038,  5154,  3276,  602;
0,  0,  12, 1952, 11328, 14192, 6194, 880;
...

Andrew Howroyd, Table of n, a(n) for n = 4..1278
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Column sums are A034187.

Cf. A059678, A059679, A059681, A059684.

T(n,k) = 0 for n > 4*k. - Andrew Howroyd, Oct 02 2017

Terms a(32) and beyond from Andrew Howroyd, Oct 02 2017

A059678 Triangle T(n,k) giving number of fixed 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).

Original entry on oeis.org

1, 0, 4, 0, 1, 8, 0, 0, 6, 12, 0, 0, 1, 18, 16, 0, 0, 0, 8, 38, 20, 0, 0, 0, 1, 32, 66, 24, 0, 0, 0, 0, 10, 88, 102, 28, 0, 0, 0, 0, 1, 50, 192, 146, 32, 0, 0, 0, 0, 0, 12, 170, 360, 198, 36, 0, 0, 0, 0, 0, 1, 72, 450, 608, 258, 40, 0, 0, 0, 0, 0, 0, 14, 292, 1002, 952, 326, 44, 0, 0, 0

Offset: 2

N. J. A. Sloane, Feb 05 2001

			Triangle begins:
1;
0, 4;
0, 1, 8;
0, 0, 6, 12;
0, 0, 1, 18, 16;
0, 0, 0,  8, 38, 20;
0, 0, 0,  1, 32, 66, 24;
...

Andrew Howroyd, Table of n, a(n) for n = 2..1276
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Column sums are A034182.

Cf. A059679, A059680, A059681, A059682.

with(combinat): for n from 2 to 30 do for k from 1 to n-1 do printf(`%d,`,sum(binomial(n-k+1, 2*k-n-v)*binomial(n-k+v, n-k), v=0..k) ) od:od:

t[n_, k_] := Sum[Binomial[n-k+1, 2*k-n-v]*Binomial[n-k+v, n-k], {v, 0, k}]; Table[t[n, k], {n, 2, 15}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Dec 20 2013 *)

T(n, k) = Sum_v C(n-k+1, 2*k-n-v)*C(n-k+v, n-k).
G.f. (1+x*y)^2/(1-x*y)*1/((1-x*y)-(1+x*y)*x^2*y). - Christopher Hanusa (chanusa(AT)math.washington.edu), Sep 22 2004
T(n,k) = 0 for n > 2*k. - Andrew Howroyd, Oct 02 2017

More terms from James Sellers, Feb 06 2001

A059681 Triangle T(n,k) giving number of fixed 5 X k polyominoes with n cells (n >= 5, 1<=k<=n-4).

Original entry on oeis.org

1, 0, 16, 0, 38, 83, 0, 32, 376, 230, 0, 10, 784, 1526, 497, 0, 1, 987, 5154, 4180, 932, 0, 0, 778, 11328, 18944, 9458, 1591, 0, 0, 370, 17598, 58665, 52488, 18936, 2538, 0, 0, 101, 19912, 135325, 204466, 123652, 34726, 3845, 0, 0, 15, 16440, 241550, 611859

Offset: 5

N. J. A. Sloane, Feb 05 2001

			Triangle starts:
1;
0, 16;
0, 38,  83;
0, 32, 376,   230;
0, 10, 784,  1526,   497;
0,  1, 987,  5154,  4180,  932;
0,  0, 778, 11328, 18944, 9458, 1591;
...

Andrew Howroyd, Table of n, a(n) for n = 5..1279
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Column sums are row 5 of A292357.

Cf. A059678, A059679, A059680.

T(n,k) = 0 for n > 5*k. - Andrew Howroyd, Oct 02 2017

a(24) corrected and terms a(26) and beyond from Andrew Howroyd, Oct 02 2017

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A034184 Not necessarily symmetric n X 3 crossword puzzle grids.

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A059680 Triangle T(n,k) read by rows giving number of fixed 4 X k polyominoes with n cells (n >= 4, 1<=k<=n-3).

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A059678 Triangle T(n,k) giving number of fixed 2 X k polyominoes with n cells (n >= 2, 1<=k<=n-1).

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A059681 Triangle T(n,k) giving number of fixed 5 X k polyominoes with n cells (n >= 5, 1<=k<=n-4).

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Formula

Extensions