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A133300
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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.
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1
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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, 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, 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
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OFFSET
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1,8
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LINKS
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B. E. Tenner, Spotlight Tiling, Ann. Comb. 14 (2011), pp. 553-568; arXiv:0711.1819 [math.CO] see p. 1.
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FORMULA
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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.
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EXAMPLE
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Using any vertical dominoes and the horizontal domino |*| |, there are two ways to tile the checkerboard
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MAPLE
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S:= (n, m)-> `if`(irem(n*m, 2)=1, 0, `if`(irem(n, 2)=0, 1,
floor((n+1)/2)^(m/2))):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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