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 A003435 Number of directed Hamiltonian circuits on n-octahedron with a marked starting node. (Formerly M4578) 3
 8, 192, 11904, 1125120, 153262080, 28507207680, 6951513784320, 2153151603671040, 826060810479206400, 384600188992919961600, 213656089636192754073600, 139620366072628402087526400, 106033731334825319789808844800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Also called the relaxed menage problem (cf. A000179). These are labeled and the order and starting point matter. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 2..100 Bogart, Kenneth P. and Doyle, Peter G., Nonsexist solution of the menage problem, Amer. Math. Monthly 93 (1986), no. 7, 514-519. D. Singmaster, Enumerating unlabeled Hamiltonian circuts, Preprint (1974). D. Singmaster, Hamiltonian circuits on the n-dimensional octahedron, J. Combinatorial Theory Ser. B 19 (1975), no. 1, 1-4. D. Singmaster, Letter to N. J. A. Sloane, May 1975 Eric Weisstein's World of Mathematics, Cocktail Party Graph FORMULA For n >= 2, a(n) = Sum_{k=0..n}(-1)^k*binomial(n, k)*((2*n)/(2*n-k))*2^k*(2*n-k)!. Conjecture: a(n) +(-4*n^2 + 2*n - 5)*a(n-1) + 2*(n-1)*(4*n-17)*a(n-2) + 12*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Oct 02 2013 Recurrence: (2*n-3)*a(n) = 2*n*(4*n^2 - 8*n + 5)*a(n-1) + 4*(n-1)*n*(2*n-1)*a(n-2). - Vaclav Kotesovec, Feb 12 2014 a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / exp(2*n+1). - Vaclav Kotesovec, Feb 12 2014 a(n) = -(-2)^(n+1)*n!*hypergeom([n, -n], [], 1/2). - Peter Luschny, Nov 10 2016 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. MAPLE A003435 := n->add((-1)^k*binomial(n, k)*((2*n)/(2*n-k))*2^k*(2*n-k)!, k=0..n); MATHEMATICA a[n_] := 2^n*n!*(2n-1)!!*Hypergeometric1F1[-n, 1-2n, -2]; Table[ a[n], {n, 2, 14}] (* Jean-François Alcover, Nov 04 2011 *) 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 CROSSREFS Cf. A003436, A003437, A129348. Sequence in context: A268095 A058873 A052734 * A071303 A128406 A265269 Adjacent sequences:  A003432 A003433 A003434 * A003436 A003437 A003438 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS Name made more precise by Andrew Howroyd, May 14 2017 STATUS approved

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Last modified December 18 05:48 EST 2018. Contains 318215 sequences. (Running on oeis4.)