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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.

%I A292357 #52 Aug 05 2024 09:55:47
%S A292357 1,1,1,1,5,1,1,15,15,1,1,39,111,39,1,1,97,649,649,97,1,1,237,3495,
%T A292357 7943,3495,237,1,1,575,18189,86995,86995,18189,575,1,1,1391,93231,
%U A292357 910667,1890403,910667,93231,1391,1
%N A292357 Array read by antidiagonals: T(m,n) is the number of fixed polyominoes that have a width of m and height of n.
%C A292357 Equivalently, the number of m X n binary arrays with all 1's connected and at least one 1 on each edge.
%H A292357 Andrew Howroyd, <a href="/A292357/b292357.txt">Table of n, a(n) for n = 1..435</a>
%H A292357 Andrew Howroyd, <a href="/A292357/a292357.txt">Fixed polyominoes by width, height and number of cells</a>
%H A292357 Louis Marin, <a href="https://arxiv.org/abs/2406.16413">Counting Polyominoes in a Rectangle b X h</a>, arXiv:2406.16413 [cs.DM], 2024. See p. 145.
%H A292357 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>
%F A292357 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).
%e A292357 Array begins:
%e A292357 ===============================================================
%e A292357 m\n| 1   2     3       4         5           6             7
%e A292357 ---|-----------------------------------------------------------
%e A292357 1  | 1   1     1       1         1           1             1...
%e A292357 2  | 1   5    15      39        97         237           575...
%e A292357 3  | 1  15   111     649      3495       18189         93231...
%e A292357 4  | 1  39   649    7943     86995      910667       9339937...
%e A292357 5  | 1  97  3495   86995   1890403    38916067     782256643...
%e A292357 6  | 1 237 18189  910667  38916067  1562052227   61025668579...
%e A292357 7  | 1 575 93231 9339937 782256643 61025668579 4617328590967...
%e A292357 ...
%e A292357 T(2,2) = 5 counts 4 3-ominoes of shape 2x2 and 1 4-omino of shape 2x2.
%e A292357 T(3,2) = 15 counts 8 4-ominoes of shape 3x2, 6 5-ominoes of shape 3x2, and 1 6-omino of shape 3x2.
%e A292357 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.
%t A292357 A287151 = Import["https://oeis.org/A287151/b287151.txt", "Table"][[All, 2]];
%t A292357 imax = Length[A287151];
%t A292357 mmax = Sqrt[2 imax] // Ceiling;
%t A292357 Clear[V]; VV = Table[V[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten;
%t A292357 Do[Evaluate[VV[[i]]] = A287151[[i]], {i, 1, imax}];
%t A292357 V[0, _] = V[_, 0] = 0;
%t A292357 T[m_, n_] := If[m == 1 || n == 1, 1, U[m, n] - 2 U[m, n-1] + U[m, n-2]];
%t A292357 U[m_, n_] := V[m, n] - 2 V[m-1, n] + V[m-2, n];
%t A292357 Table[T[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten // Take[#, imax]& (* _Jean-François Alcover_, Sep 22 2019 *)
%Y A292357 Rows 2..4 are A034182, A034184, A034187.
%Y A292357 Main diagonal is A268404.
%Y A292357 Cf. A268371 (nonequivalent), A287151, A308359.
%K A292357 nonn,tabl
%O A292357 1,5
%A A292357 _Andrew Howroyd_, Oct 02 2017