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.
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
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
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Oct 02 2017
STATUS
approved