A059679 -id:A059679

Search: a059679 -id:a059679

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page 1

A034184

Not necessarily symmetric n X 3 crossword puzzle grids.

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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.

FORMULA

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

CROSSREFS

Row 3 of A292357.

Column sums of A059679.

Cf. A001333, A034182, A034184, A034185, A034186, A034187.

KEYWORD

nonn

AUTHOR

Erich Friedman

STATUS

approved

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).

+10
6

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

4,3

LINKS

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.

FORMULA

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

EXAMPLE

Triangle starts:

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;

...

CROSSREFS

Column sums are A034187.

Cf. A059678, A059679, A059681, A059684.

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane, Feb 05 2001

EXTENSIONS

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

STATUS

approved

A059678

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

+10
5

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

2,3

LINKS

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.

FORMULA

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

EXAMPLE

Triangle begins:

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;

...

MAPLE

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:

MATHEMATICA

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 *)

CROSSREFS

Column sums are A034182.

Cf. A059679, A059680, A059681, A059682.

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane, Feb 05 2001

EXTENSIONS

More terms from James A. Sellers, Feb 06 2001

STATUS

approved

A059681

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

+10
4

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

5,3

LINKS

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.

FORMULA

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

EXAMPLE

Triangle starts:

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;

...

CROSSREFS

Column sums are row 5 of A292357.

Cf. A059678, A059679, A059680.

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane, Feb 05 2001

EXTENSIONS

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

STATUS

approved

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