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A003435
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Number of Hamiltonian circuits on n-octahedron.
(Formerly M4578)
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2
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8, 192, 11904, 1125120, 153262080, 28507207680, 6951513784320, 2153151603671040, 826060810479206400, 384600188992919961600, 213656089636192754073600, 139620366072628402087526400, 106033731334825319789808844800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Also called the relaxed menage problem (cf. A000179).
These are labeled and the order and starting point matter.
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REFERENCES
| Bogart, Kenneth P. and Doyle, Peter G., Nonsexist solution of the menage problem, Amer. Math. Monthly 93 (1986), no. 7, 514-519.
Singmaster, David, Hamiltonian circuits on the n-dimensional octahedron. J. Combinatorial Theory Ser. B 19 (1975), no. 1, 1-4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| For n >= 2, a(n) = sum((-1)^k*binomial(n, k)*((2*n)/(2*n-k))*2^k*(2*n-k)!, k=0..n).
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EXAMPLE
| n=2: label vertices of a square 1,2,3,4. Then the 8 Hamiltonian circuits are 1234, 1432, 2341, 2143, 3412, 3214, 4123, 4321; so a(2) = 8.
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MAPLE
| A003435 := n->add((-1)^k*binomial(n, k)*((2*n)/(2*n-k))*2^k*(2*n-k)!, k=0..n);
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MATHEMATICA
| a[n_] := 2^n*n!*(2n-1)!!*Hypergeometric1F1[-n, 1-2n, -2]; Table[ a[n], {n, 2, 14}] (* From Jean-François Alcover, Nov 04 2011 *)
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PROG
| (PARI) a(n)=sum(k=0, n, (-1)^k*binomial(n, k)*((2*n)/(2*n-k))*2^k*(2*n-k)!) \\ Charles R Greathouse IV, Nov 04 2011
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CROSSREFS
| Cf. A003436, A003437.
Sequence in context: A058873 A174091 A052734 * A071303 A128406 A003956
Adjacent sequences: A003432 A003433 A003434 * A003436 A003437 A003438
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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