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Search: id:A102080
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| A102080 |
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Number of matchings in the C_n X P_2 graph (C_n is the cycle graph on n vertices and P_2 is the path graph on 2 vertices). |
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+0
2
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| 12, 32, 108, 342, 1104, 3544, 11396, 36626, 117732, 378424, 1216380, 3909830, 12567448, 40395792, 129844996, 417363330, 1341539196, 4312135920, 13860583628, 44552347606, 143205490528, 460308235560, 1479577849604
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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a(n)=sum of row n in A102079.
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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.(23) and Table IV).
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FORMULA
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G.f.= 2z^2*(6+4z-2z^2-z^3)/[(z+1)(z^3-z^2-3*z+1)]. a(n)=2a(n-1)+4a(n-2)-a(n-4) (n>=6).
A033505(n) - 7*A033505(n-2) - (-1)^n. - Ralf Stephan, May 17 2007
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EXAMPLE
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Example: a(3)=32 because in the graph with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,BC, A'B',A'C',B'C',AA',BB',CC'} we have the following matchings:
(i) the empty set (1 matching), (ii) any edge (9 matchings), (iii) any two edges from the set {AA',BB',CC'} (3 matchings), (iv) the members of the Cartesian product of {AB,AC,BC}and {A'B',A'C',B'C'} (9 matchings), (v) {AA',BC}, {AA',B'C'}and four more obtained by circular permutations (6 matchings), (vi) {AA',BC,B'C'} and two more obtained by circular permutations (3 matchings), (vii) {AA',BB',CC'} (1 matching).
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MAPLE
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a[2]:=12: a[3]:=32: a[4]:=108: a[5]:=342: for n from 6 to 30 do a[n]:=2*a[n-1]+4*a[n-2]-a[n-4] od:seq(a[n], n=2..27);
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CROSSREFS
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Cf. A102079, A102081.
Sequence in context: A081268 A068381 A143238 this_sequence A102091 A045669 A045660
Adjacent sequences: A102077 A102078 A102079 this_sequence A102081 A102082 A102083
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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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