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A006069 Number of directed Hamiltonian cycles (or Gray codes) on n-cube with a marked starting node.

%I A006069 M1903 #46 Jul 08 2025 16:41:06
%S A006069 2,8,96,43008,58018928640,4587291356489073135452160
%N A006069 Number of directed Hamiltonian cycles (or Gray codes) on n-cube with a marked starting node.
%C A006069 More precisely, 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.
%D A006069 M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 24.
%D A006069 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006069 H. Haanpaa and Patric R. J. Östergård, <a href="http://dx.doi.org/10.1090/S0025-5718-2013-02741-X">Counting Hamiltonian cycles in bipartite graphs</a>, Math. Comp. 83 (2014), 979-995.
%H A006069 Michel Deza and Roman Shklyar, <a href="http://arxiv.org/abs/1003.4391">Enumeration of Hamiltonian Cycles in 6-cube</a>, arXiv:1003.4391 [cs.DM], 2010. [There may be errors - see Haanpaa and Ostergard, 2012]
%H A006069 D. Sensarma, S. S. Sarma, <a href="https://doi.org/10.15623/IJRET.2014.0303121">GMDES: A graph based modified Data Encryption Standard algorithm with enhanced security</a>, IJRET: International Journal of Research in Engineering and Technology 03:03 (2014), 653-660. See Section 2.2.
%H A006069 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>
%H A006069 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>
%F A006069 a(n) = A003042(n)*2^n. - _Max Alekseyev_, Jun 15 2006
%e A006069 a(1) = 2: we have 1,2 or 2,1.
%e A006069 a(2) = 8: label the nodes 1, 2, ..., 4. Then the 8 possibilities are 1,2,3,4; 1,4,3,2; 2,3,4,1; 2,1,4,3; etc.
%Y A006069 Cf. A003042, A006070, A091299, A091302, A159344.
%K A006069 nonn,more
%O A006069 1,1
%A A006069 _N. J. A. Sloane_
%E A006069 a(5) corrected by Jonathan Cross (jcross(AT)wcox.com), Oct 10 2001
%E A006069 Definition corrected by _Max Alekseyev_, Jun 15 2006
%E A006069 a(6) from Michel Deza, Mar 28 2010
%E A006069 a(6) corrected by Haanpaa and Östergård, 2012. - _N. J. A. Sloane_, Sep 06 2012