A003043 Number of Hamiltonian paths (or Gray codes) on n-cube with a marked starting node. (Formerly M2112)

1, 2, 18, 5712, 5859364320

Offset

1,2

Comments

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.

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.

References

M. Gardner, Mathematical Games, Sci. Amer. Vol. 228 (No. 4, Apr. 1973), p. 111.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..5.

Vladimir Shevelev, Combinatorial minors of matrix functions and their applications, arXiv:1105.3154 [math.CO], 2011-2014.

Vladimir Shevelev, Combinatorial minors of matrix functions and their applications, Zesz. Nauk. PS., Mat. Stosow., Zeszyt 4, pp. 5-16. (2014).

Formula

a(n) = A091299(n)/2^n.

Crossrefs

Cf. A091299, A006069, A006070, A003042, A066037, A091302, A179926.

Sequence in context: A060598 A055687 A006262 * A059783 A309972 A208056

Adjacent sequences: A003040 A003041 A003042 * A003044 A003045 A003046

Keyword

nonn,hard,more

Author

N. J. A. Sloane

Extensions

a(5) (from A091299) from Max Alekseyev, Jul 09 2006

Alternative description added by Jeffrey Shallit, Feb 02 2013

Status

approved