A059678 -id:A059678 - cp's OEIS frontend

A034182 Number of not-necessarily-symmetric n X 2 crossword puzzle grids.

1, 5, 15, 39, 97, 237, 575, 1391, 3361, 8117, 19599, 47319, 114241, 275805, 665855, 1607519, 3880897, 9369317, 22619535, 54608391, 131836321, 318281037, 768398399, 1855077839, 4478554081, 10812186005, 26102926095, 63018038199, 152139002497, 367296043197

Offset: 1

Erich Friedman

n X 2 binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002

Define a triangle with T(n,1) = T(n,n) = n*(n-1) + 1, n>=1, and its interior terms via T(r,c) = T(r-1,c) + T(r-1,c-1)+ T(r-2,c-1), 2<=cJ. M. Bergot, Mar 16 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1)

Row 2 of A292357.
Column sums of A059678.
Cf. A001333, A034184, A034187, A052542 (first differences).

a034182 n = a034182_list !! (n-1)
a034182_list = 1 : 5 : (map (+ 4) $
   zipWith (+) a034182_list (map (* 2) $ tail a034182_list))
-- Reinhard Zumkeller, May 23 2013

{1}~Join~NestList[{#2, 2 #2 + #1 + 4} & @@ # &, {1, 5}, 28][[All, -1]] (* Michael De Vlieger, Oct 02 2017 *)

a(n) = 2a(n-1) + a(n-2) + 4.
(1 + 5x + 15x^2 + ...) = (1 + 2x + 2x^2 + ...) * (1 + 3x + 7x^2 + ...), convolution of A040000 and left-shifted A001333.
a(n) = (-4 + (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n))/2. G.f.: x*(1+x)^2/((1-x)*(1 - 2*x - x^2)). - Colin Barker, May 22 2012
a(n) = A001333(n+1)-2. - R. J. Mathar, Mar 28 2013
a(n) = A048739(n-3) +2*A048739(n-2) +A048739(n-1). - R. J. Mathar, Jun 15 2020

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

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

cp's OEIS Frontend

A034182 Number of not-necessarily-symmetric n X 2 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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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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Extensions