A034187 -id:A034187 - 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

A292357 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 39, 111, 39, 1, 1, 97, 649, 649, 97, 1, 1, 237, 3495, 7943, 3495, 237, 1, 1, 575, 18189, 86995, 86995, 18189, 575, 1, 1, 1391, 93231, 910667, 1890403, 910667, 93231, 1391, 1

Offset: 1

Andrew Howroyd, Oct 02 2017

Equivalently, the number of m X n binary arrays with all 1's connected and at least one 1 on each edge.

			Array begins:
===============================================================
m\n| 1   2     3       4         5           6             7
---|-----------------------------------------------------------
1  | 1   1     1       1         1           1             1...
2  | 1   5    15      39        97         237           575...
3  | 1  15   111     649      3495       18189         93231...
4  | 1  39   649    7943     86995      910667       9339937...
5  | 1  97  3495   86995   1890403    38916067     782256643...
6  | 1 237 18189  910667  38916067  1562052227   61025668579...
7  | 1 575 93231 9339937 782256643 61025668579 4617328590967...
...
T(2,2) = 5 counts 4 3-ominoes of shape 2x2 and 1 4-omino of shape 2x2.
T(3,2) = 15 counts 8 4-ominoes of shape 3x2, 6 5-ominoes of shape 3x2, and 1 6-omino of shape 3x2.
T(4,2) = 39 counts 12 5-ominoes of shape 4x2, 18 6-ominoes of shape 4x2, 8 7-ominoes of shape 4x2, and 1 8-omino of shape 4x2.

Andrew Howroyd, Table of n, a(n) for n = 1..435
Andrew Howroyd, Fixed polyominoes by width, height and number of cells
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.
Index entries for sequences related to polyominoes

Rows 2..4 are A034182, A034184, A034187.
Main diagonal is A268404.
Cf. A268371 (nonequivalent), A287151, A308359.

A287151 = Import["https://oeis.org/A287151/b287151.txt", "Table"][[All, 2]];
imax = Length[A287151];
mmax = Sqrt[2 imax] // Ceiling;
Clear[V]; VV = Table[V[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten;
Do[Evaluate[VV[[i]]] = A287151[[i]], {i, 1, imax}];
V[0, ] = V[, 0] = 0;
T[m_, n_] := If[m == 1 || n == 1, 1, U[m, n] - 2 U[m, n-1] + U[m, n-2]];
U[m_, n_] := V[m, n] - 2 V[m-1, n] + V[m-2, n];
Table[T[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten // Take[#, imax]& (* Jean-François Alcover, Sep 22 2019 *)

T(m, n) = U(m, n) - 2*U(m, n-1) + U(m, n-2) where U(m, n) = V(m, n) - 2*V(m-1, n) + V(m-2, n) and V(m, n) = A287151(m, n).

A034184 Not necessarily symmetric n X 3 crossword puzzle grids.

Original entry on oeis.org

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

A034185 Number of symmetric n X 3 crossword puzzle grids.

Original entry on oeis.org

1, 3, 11, 21, 61, 111, 309, 565, 1563, 2859, 7907, 14459, 39991, 73117, 202227, 369729, 1022587, 1869581, 5170829, 9453757, 26146909, 47804041, 132214959, 241726811, 668560673, 1222320359, 3380656559, 6180808253, 17094691953, 31253992007, 86441342981

Offset: 1

Erich Friedman

nonn

Rules: all white squares are edge connected; at least 1 white square on every edge of grid; symmetric ones have 180-degree rotational symmetry.

Sean A. Irvine, Java program (github)

Cf. A001333, A034184, A034186, A034187.

a(23)-a(31) from Sean A. Irvine, Aug 07 2020

A034186 Number of symmetric n X 4 crossword puzzle grids.

Original entry on oeis.org

1, 3, 21, 47, 227, 499, 2321, 5087, 23429, 51263, 235881, 515691, 2372511, 5185609, 23856215, 52139887, 239863701, 524238555, 2411688233, 5270900439, 24248031407, 52995614649

Offset: 1

Erich Friedman

nonn

Cf. A001333, A034184-A034187.

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

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A292357 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n.

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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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A034185 Number of symmetric n X 3 crossword puzzle grids.

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A034186 Number of symmetric n X 4 crossword puzzle grids.

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