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