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

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

1,5

Comments

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

Links

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

Formula

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).

Example

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.

Mathematica

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 *)

Crossrefs

Rows 2..4 are A034182, A034184, A034187.

Main diagonal is A268404.

Cf. A268371 (nonequivalent), A287151, A308359.

Sequence in context: A157147 A347973 A232103 * A156920 A174044 A174159

Adjacent sequences: A292354 A292355 A292356 * A292358 A292359 A292360

Keyword

nonn,tabl

Author

Andrew Howroyd, Oct 02 2017

Status

approved