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%I #14 Jan 30 2016 02:07:26
%S 0,1,1,0,1,0,1,2,1,1,0,1,0,1,0,1,3,1,4,1,1,0,1,0,1,0,1,0,1,4,1,9,1,8,
%T 1,1,0,1,0,1,0,1,0,1,0,1,5,1,16,1,27,1,16,1,1,0,1,0,1,0,1,0,1,0,1,0,1,
%U 6,1,25,1,64,1,81,1,32,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,7,1,36,1,125,1,256,1,243,1,64,1,1
%N Square array read along upward antidiagonals. S(n,m) is the number of domino tilings of an n-row and m-column checkerboard with a black upper-left square, where any vertical dominoes are allowed and horizontal dominoes must be placed so that the black square is on the left.
%H Alois P. Heinz, <a href="/A133300/b133300.txt">Rows n = 1..141, flattened</a>
%H B. E. Tenner, <a href="http://arxiv.org/abs/0711.1819">Spotlight Tiling</a>, Ann. Comb. 14 (2011), pp. 553-568; arXiv:0711.1819 [math.CO] see p. 1.
%F S(n,m) = 0 if m and n are odd, 1 if n is even, or [(n+1)/2]^(m/2) if n is odd and m is even.
%e Using any vertical dominoes and the horizontal domino |*| |, there are two ways to tile the checkerboard
%e -----
%e |*| |
%e -----
%e | |*|
%e -----
%e |*| |
%e -----
%p S:= (n, m)-> `if`(irem(n*m, 2)=1, 0, `if`(irem(n, 2)=0, 1,
%p floor((n+1)/2)^(m/2))):
%p seq(seq(S(1+d-m, m), m=1..d), d=1..14); # _Alois P. Heinz_, Nov 10 2013
%t S[n_, m_] := If[Mod[n*m, 2]==1, 0, If[Mod[n, 2]==0, 1, Floor[(n+1)/2]^(m/2) ]]; Table[S[1+d-m, m], {d, 1, 14}, {m, 1, d}] // Flatten (* _Jean-François Alcover_, Jan 30 2016, after _Alois P. Heinz_ *)
%K nonn,tabl
%O 1,8
%A _Bridget Tenner_, Oct 17 2007
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