A133300 - OEIS

A133300

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.

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

1,8

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

B. E. Tenner, Spotlight Tiling, Ann. Comb. 14 (2011), pp. 553-568; arXiv:0711.1819 [math.CO] see p. 1.

FORMULA

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.

EXAMPLE

Using any vertical dominoes and the horizontal domino |*| |, there are two ways to tile the checkerboard

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|*| |

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| |*|

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|*| |

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MAPLE

S:= (n, m)-> `if`(irem(n*m, 2)=1, 0, `if`(irem(n, 2)=0, 1,

floor((n+1)/2)^(m/2))):

seq(seq(S(1+d-m, m), m=1..d), d=1..14); # Alois P. Heinz, Nov 10 2013

MATHEMATICA

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 *)

CROSSREFS

Sequence in context: A092510 A117208 A276004 * A337101 A178779 A144451

Adjacent sequences: A133297 A133298 A133299 * A133301 A133302 A133303

KEYWORD

nonn,tabl

AUTHOR

Bridget Tenner, Oct 17 2007

STATUS

approved