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 A066037 Number of (undirected) Hamiltonian cycles in the binary n-cube, or the number of cyclic n-bit Gray codes. 10
 1, 1, 6, 1344, 906545760, 35838213722570883870720 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS This is the number of ways of making a list of the 2^n nodes of the n-cube, with a distinguished starting position and a direction, such that each node is adjacent to the previous one and the last node is adjacent to the first; and then dividing the total by 2^(n+1) because the starting node and the direction do not really matter. The number is a multiple of n!/2 since any directed cycle starting from 0^n induces a permutation on the n bits, namely the order in which they first get set to 1. LINKS Michel Deza and Roman Shklyar, Enumeration of Hamiltonian Cycles in 6-cube, arXiv:1003.4391 [cs.DM], 2010. [There may be errors - see Haanpaa and Ostergard, 2012] R. J. Douglas, Bounds on the number of Hamiltonian circuits in the n-cube, Discrete Mathematics, 17 (1977), 143-146. Harri Haanpaa and Patric R. J. Östergård, Counting Hamiltonian cycles in bipartite graphs, Math. Comp. 83 (2014), 979-995. Harary, Hayes, and Wu, A survey of the theory of hypercube graphs, Computers and Mathematics with Applications, 15(4), 1988, 277-289. Eric Weisstein's World of Mathematics, Hamiltonian Cycle Eric Weisstein's World of Mathematics, Hypercube Graph EXAMPLE The 2-cube has a single cycle consisting of all 4 edges. MATHEMATICA Prepend[Table[Length[FindHamiltonianCycle[HypercubeGraph[n], All]], {n, 2, 4}], 1] (* Eric W. Weisstein, Apr 01 2017 *) CROSSREFS Equals A006069/2^(n+1) and A003042/2. Cf. A006070, A091299, A003043, A091302. Cf. A236602 (superset). - Stanislav Sykora, Feb 01 2014 Sequence in context: A055306 A203331 A172625 * A279924 A307871 A195646 Adjacent sequences:  A066034 A066035 A066036 * A066038 A066039 A066040 KEYWORD nonn,nice,more AUTHOR John Tromp, Dec 12 2001 EXTENSIONS a(6) from Michel Deza, Mar 28 2010 a(6) corrected by Haanpaa and Östergård, 2012. - N. J. A. Sloane, Sep 06 2012 Name clarified by Eric W. Weisstein, May 06 2019 STATUS approved

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)