A007030 Non-Hamiltonian simplicial polyhedra with n nodes. (Formerly M2152)

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 30, 239, 2369, 22039, 205663, 1879665, 16999932, 152227187, 1353996482

Offset

1,12

Comments

a(18) = 1879665 was conjectured by Dillencourt and verified by direct computation by Sean A. Irvine, Sep 26 2017.

By Steinitz's theorem non-Hamiltonian simplicial polyhedra correspond to non-Hamiltonian maximal planar graphs. - William P. Orrick, Feb 25 2021

References

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties. Tech. Rep. 92-91, Info. and Comp. Sci. Dept., Univ. Calif. Irvine, 1992.

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..21.

M. B. Dillencourt, Polyhedra of small orders and their Hamiltonian properties, Journal of Combinatorial Theory, Series B, Volume 66, Issue 1, January 1996, Pages 87-122.

Eric Weisstein's World of Mathematics, Polyhedral Graph

Wikimedia, Goldner-Harary graphs, additional images of the graph and related simplicial polyhedron created by David Eppstein and Richard J. Mathar. - William P. Orrick, Feb 25 2021

Wikipedia, Goldner-Harary graph

Formula

a(n) = A000109(n) - A115340(n-2). - William P. Orrick, Feb 20 2021

Example

The unique non-Hamiltonian maximal planar graph of 11 vertices is the Goldner-Harary graph. A corresponding simplicial polyhedron can be obtained by attaching a tetrahedron to each of the six faces of a triangular bipyramid. - William P. Orrick, Feb 25 2021

Crossrefs

Cf. A000109, A115340.

Sequence in context: A089288 A232602 A154413 * A157054 A092355 A215237

Adjacent sequences: A007027 A007028 A007029 * A007031 A007032 A007033

Keyword

nonn,hard,more

Author

N. J. A. Sloane.

Extensions

a(18) from Sean A. Irvine, Sep 26 2017

a(19)-a(21) using new formula by William P. Orrick, Feb 20 2021

Status

approved