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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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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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-Fran├žois Alcover, Jan 30 2016, after Alois P. Heinz *)

CROSSREFS

Sequence in context: A092510 A117208 A276004 * A178779 A144451 A259287

Adjacent sequences:  A133297 A133298 A133299 * A133301 A133302 A133303

KEYWORD

nonn,tabl

AUTHOR

Bridget Tenner, Oct 17 2007

STATUS

approved

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Last modified September 21 01:47 EDT 2019. Contains 327252 sequences. (Running on oeis4.)