A028475 Total number of Hamiltonian cycles avoiding the root-edge in rooted cubic bipartite planar maps with 2n nodes.

1, 4, 20, 114, 712, 4760, 33532, 246146, 1867556, 14557064, 116038672, 942597638, 7781117632, 65131605840, 551825148660, 4725380142050, 40848069782932, 356094155836640, 3127831256055624, 27662285924478844, 246164019830290392, 2203001550262470312, 19817596934324929372

Offset: 1

Author

Valery A. Liskovets, Apr 29 2002

Keywords

• nonn
• hard

Comments

An algorithm for calculating these numbers is known. 2*a(n) can be interpreted as the number of pairs of non-intersecting arch configurations (over and under a straight line) connecting 2n points in the line, where all points are marked + and - alternately, every point belongs to a unique arch and the ends of every arch have different signs.

Examples

			n=2. There are 3 rooted cubic bipartite planar maps with 4 nodes: a quadrangular with two non-adjacent edges doubled (parallel), where one vertex and any of the edges incident to it are taken as the root. No Hamiltonian cycle can avoid the sole edge incident to the root-vertex. For the other two rootings, there are 4 root-edge avoiding Hamiltonian cycles. So a(2)=4.
		

Links

• Olivier Golinelli, Table of n, a(n) for n = 1..34
• E. Guitter, C. Kristjansen and J. L. Nielsen, Hamiltonian cycles on random Eulerian triangulations, arXiv:cond-mat/9811289 [cond-mat.stat-mech], 1998; Nucl.Phys. B546 (1999), No.3, 731-750.
• James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.

Crossrefs

Cf. A000356, A003122, A007084, A116456.

Formula

a(n) = A116456(n) / 2. - Sean A. Irvine, Feb 01 2020

Extensions

a(21)-a(32) from Cyril Banderier, Nov 06 2022