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A099390
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Array T(m,n) read by antidiagonals: number of domino tilings of the m X n grid (m>=1, n>=1).
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21
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0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 5, 0, 5, 0, 1, 8, 11, 11, 8, 1, 0, 13, 0, 36, 0, 13, 0, 1, 21, 41, 95, 95, 41, 21, 1, 0, 34, 0, 281, 0, 281, 0, 34, 0, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 0, 89, 0, 2245, 0, 6728, 0, 2245, 0, 89, 0, 1, 144, 571, 6336, 14824, 31529, 31529, 14824, 6336, 571, 144, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 1..1035
F. Ardila and R. P. Stanley, Tilings
F. Faase, Counting Hamilton cycles in product graphs
F. Faase, Results from the counting program
P. E. John and H. Sachs, On a strange observation in the theory of the dimer problem
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
J. Propp, Dimers and Dominoes
Index entries for sequences related to dominoes
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FORMULA
| If m, n even then T(m, n) = Prod(j=1..m/2, Prod(k=1..n/2, 4*cos(j*Pi/(m+1))^2 + 4*cos(k*Pi/(n+1))^2)).
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EXAMPLE
| 0, 1, 0, 1, 0, 1, ...
1, 2, 3, 5, 8, 13, ...
0, 3, 0, 11, 0, 41, ...
1, 5, 11, 36, 95, 281, ...
0, 8, 0, 95, 0, 1183, ...
1, 13, 41, 281, 1183, 6728, ...
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MATHEMATICA
| t[m_, n_] := Product[2*(2 + Cos[2j*Pi/(m+1)] + Cos[2k*Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}]; t[_?OddQ, _?OddQ] = 0; Flatten[ Table[ FullSimplify[ t[m-n+1, n]], {m, 1, 12}, {n, 1, m}]](* From Jean-François Alcover, Nov 25 2011 *)
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CROSSREFS
| See A187596 for another version (with m >= 0, n >= 0). See A187616 for a triangular version. See also A187617, A187618.
See also A004003 for more literature on the dimer problem.
Rows 2-12 (without zeros) are A000045, A001835, A005178, A003775, A028468, A028469, A028470, A028471, A028472, A028473, A028474.
Main diagonal is A004003.
Cf. A103997, A103999.
Sequence in context: A103438 A167279 A068920 * A124031 A049600 A004542
Adjacent sequences: A099387 A099388 A099389 * A099391 A099392 A099393
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KEYWORD
| tabl,nonn
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AUTHOR
| Ralf Stephan, Oct 16 2004
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EXTENSIONS
| Corrected broken URL's. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 06 2009
Fixed old link and added link to results page. - Frans Faase (faase009(AT)planet.nl), Feb 04 2009
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