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%I M5295 #21 Dec 25 2021 14:29:33
%S 0,0,48,48384,129480729600
%N Number of Hamiltonian paths on n-cube which are strictly not cycles.
%C Number of Gray codes of length n which strictly do not close.
%C 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 not adjacent to the first.
%D M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments. Freeman, NY, 1986, p. 24.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HypercubeGraph.html">Hypercube Graph</a>
%F a(n) = A091299(n) - A006069(n). - _Andrew Howroyd_, Dec 25 2021
%e There are no such paths for n=1 or n=2 (the square). For n = 3 every path has to end at the node of the cube that is diametrically opposite to the start. There are 16 choices for the start and for each start there are 3 Hamiltonian paths that end at the opposite node, so a(3) = 3*16 = 48.
%Y Cf. A006069, A091299.
%K nonn,hard,more
%O 1,3
%A _N. J. A. Sloane_
%E a(5) from Greg Barton (greg_barton(AT)yahoo.com), May 24 2004
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