A059680 -id:A059680 - cp's OEIS frontend

A034187 Not necessarily symmetric n X 4 crossword puzzle grids.

1, 39, 649, 7943, 86995, 910667, 9339937, 94844591, 958363411, 9659847433, 97245624749, 978360244839, 9839915415611, 98949930968385, 994959069405031, 10004090931544495, 100586881489055547, 1011348141567934109

Offset: 1

Erich Friedman

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 4 of A292357.
Column sums of A059680.
Cf. A001333, A034184, A034185, A034186, A034187.

Empirical: a(n) = 24*a(n-1) - 218*a(n-2) + 1009*a(n-3) - 2623*a(n-4) + 3513*a(n-5) - 142*a(n-6) - 7707*a(n-7) + 11632*a(n-8) - 4443*a(n-9) - 6736*a(n-10) + 9655*a(n-11) - 3714*a(n-12) - 1529*a(n-13) + 1550*a(n-14) - 597*a(n-15) + 1041*a(n-16) + 195*a(n-17) - 150*a(n-18) + 8*a(n-19) + a(n-20) for n > 20. - Andrew Howroyd, Oct 02 2017

Empirical: x *(-1 -15*x +69*x^2 +140*x^3 -1117*x^4 +1696*x^5 +186*x^6 -1351*x^7 -1715*x^8 +3376*x^9 -52*x^10 -1829*x^11 +905*x^12 -1748*x^13 +446*x^14 -241*x^15 -72*x^16 +35*x^17 +7*x^18 +x^19) / ( (x^2+2*x-1) *(x^6 -7*x^5 +x^4 +6*x^3 -11*x^2 +7*x -1) *(x^12 +13*x^11 -71*x^10 +7*x^9 -113*x^8 +22*x^7 +222*x^6 -210*x^5 -19*x^4 +97*x^3 -59*x^2 +15*x -1) ). - R. J. Mathar, Jun 07 2020

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

A059679 Triangle T(n,k) giving number of fixed 3 X k polyominoes with n cells (n >= 3, 1<=k<=n-2).

Original entry on oeis.org

1, 0, 8, 0, 6, 25, 0, 1, 44, 50, 0, 0, 32, 154, 83, 0, 0, 9, 212, 376, 124, 0, 0, 1, 158, 784, 750, 173, 0, 0, 0, 62, 987, 2133, 1316, 230, 0, 0, 0, 12, 778, 3802, 4803, 2114, 295, 0, 0, 0, 1, 370, 4622, 11127, 9490, 3184, 368

Offset: 3

N. J. A. Sloane, Feb 05 2001

			Triangle starts:
1;
0, 8;
0, 6, 25;
0, 1, 44,  50;
0, 0, 32, 154,  83;
0, 0,  9, 212, 376,  124;
0, 0,  1, 158, 784,  750,  173;
0, 0,  0,  62, 987, 2133, 1316, 230;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Column sums are A034184.

Cf. A059678, A059680, A059681, A059683.

T(n,k) = 0 for n > 3*k. - Andrew Howroyd, Oct 02 2017

a(8) corrected and terms a(39) and beyond from Andrew Howroyd, Oct 02 2017

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

A059684 Triangle T(n,k) giving number of 4 X k polyominoes with n cells (n >= 4, 1<=k<=n-3).

Original entry on oeis.org

1, 0, 3, 0, 6, 15, 0, 2, 39, 30, 0, 1, 59, 148, 61, 0, 0, 42, 349, 383, 97, 0, 0, 21, 519, 1304, 822, 155, 0, 0, 4, 488, 2847, 3548, 1551, 220, 0, 0, 1, 321, 4441, 10323, 8239, 2680, 313, 0, 0, 0, 122, 5008, 21995, 29442, 16821, 4327, 415, 0, 0, 0, 35, 4168, 36035, 79155, 71742, 31576

Offset: 4

N. J. A. Sloane, Feb 05 2001

Note that for k=4 (polyominoes with square bounding rectangle) these are not the free polyominoes, because Read does not apply the full symmetry group of order 8 to reduce the fixed polyominoes for d_q(n), but only the symmetry group of order 4 (excluding the 90 deg rotations). The free polyominoes with square bounding rectangles are his z_4(n) instead. - R. J. Mathar, May 12 2019

			Triangle starts:
1;
0,3;
0,6,15;
0,2,39, 30;
0,1,59,148,  61;
0,0,42,349, 383,   97;
0,0,21,519,1304,  822,  155;
0,0, 4,488,2847, 3548, 1551,  220;
0,0, 1,321,4441,10323, 8239, 2680,  313;
0,0, 0,122,5008,21995,29442,16821, 4327,415;
0,0, 0, 35,4168,36035,79155,71742,31576,...
There are T(5,2)=3 out of 12 pentominoes that fill the 4X2 shape: the L, N and Y. The F, T, V, W, X, and Z require both dimensions >= 3; the P and U would fit but not touch all sides; the I requires one dimension of 5. - _R. J. Mathar_, May 08 2019

R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Cf. A059680 (flipped or rotated considered distinct).

Changed 518 to 519 (correcting Read...) and added values for n>=11 cells. R. J. Mathar, May 12 2019

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A034187 Not necessarily symmetric n X 4 crossword puzzle grids.

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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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A059679 Triangle T(n,k) giving number of fixed 3 X k polyominoes with n cells (n >= 3, 1<=k<=n-2).

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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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A059684 Triangle T(n,k) giving number of 4 X k polyominoes with n cells (n >= 4, 1<=k<=n-3).

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