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Search: id:A102091
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| A102091 |
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Number of perfect matchings in the C_{2n} X P_3 graph (C_{2n} is the cycle graph on 2n vertices and P_3 is the path graph on 3 vertices). |
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+0
2
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| 12, 32, 108, 392, 1452, 5408, 20172, 75272, 280908, 1048352, 3912492, 14601608, 54493932, 203374112, 759002508, 2832635912, 10571541132, 39453528608, 147242573292, 549516764552, 2050824484908, 7653781175072, 28564300215372
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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a(n)=A102089(2n,3n).
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REFERENCES
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H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (53) and Table VII).
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FORMULA
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a(n)=5a(n-1)-5a(n-2)+a(n-3) with a(1)=12, a(2)=32 and a(3)=108. G.f.=4z(1-2z)(3-z)/[(1-z)(1-4z+z^2)].
A001353(n+1) - 7*A001353(n-1) + 4. - Ralf Stephan, May 17 2007
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EXAMPLE
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a(1)=12 because in the graph C_2 X P_3 with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,A'B',B'C',a,a',b,b',c,c'}, where a and a' are two edges between A and A', b and b' are two edges between B and B' and c and c' are two edges between C and C', we have the following twelve perfect matchings: eight matchings by taking one edge from each of the pairs {a,a'},{b,b'} and {c,c'}; two matchings by taking AB, A'B' and either edge from the pair {c,c'}; two matchings by taking BC, B'C' and either edge from the pair {a,a'}.
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MAPLE
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a[1]:=12: a[2]:=32: a[3]:=108: for n from 4 to 31 do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3] od:seq(a[n], n=1..25);
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CROSSREFS
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Cf. A102089.
Sequence in context: A068381 A143238 A102080 this_sequence A045669 A045660 A050690
Adjacent sequences: A102088 A102089 A102090 this_sequence A102092 A102093 A102094
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 29 2004
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