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%I M2112 #40 Jan 04 2019 04:58:38
%S 1,2,18,5712,5859364320
%N Number of Hamiltonian paths (or Gray codes) on n-cube with a marked starting node.
%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. The final node may or may not be adjacent to the first. Finally, divide by 2^n since the starting node really doesn't matter.
%C Also, the number of strings s of length 2^n - 1 over the alphabet {1,2,...,n} with the property that every contiguous subblock has some letter that appears an odd number of times.
%D M. Gardner, Mathematical Games, Sci. Amer. Vol. 228 (No. 4, Apr. 1973), p. 111.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1105.3154">Combinatorial minors of matrix functions and their applications</a>, arXiv:1105.3154 [math.CO], 2011-2014.
%H Vladimir Shevelev, <a href="https://www.math.bgu.ac.il/~shevelev/comb_meth2014.pdf">Combinatorial minors of matrix functions and their applications</a>, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014).
%F a(n) = A091299(n)/2^n.
%Y Cf. A091299, A006069, A006070, A003042, A066037, A091302, A179926.
%K nonn,hard,more
%O 1,2
%A _N. J. A. Sloane_
%E a(5) (from A091299) from _Max Alekseyev_, Jul 09 2006
%E Alternative description added by _Jeffrey Shallit_, Feb 02 2013
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