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A028475 Total number of Hamiltonian cycles avoiding the root-edge in rooted cubic bipartite planar maps with 2n nodes. 2
1, 4, 20, 114, 712, 4760, 33532, 246146, 1867556, 14557064, 116038672, 942597638, 7781117632, 65131605840, 551825148660, 4725380142050, 40848069782932, 356094155836640, 3127831256055624, 27662285924478844, 246164019830290392, 2203001550262470312, 19817596934324929372 (list; graph; refs; listen; history; text; internal format)
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.
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.
a(n) = A116456(n) / 2. - Sean A. Irvine, Feb 01 2020
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.
Sequence in context: A367237 A211248 A346414 * A128327 A320615 A316298
Valery A. Liskovets, Apr 29 2002
a(21)-a(32) from Cyril Banderier, Nov 06 2022

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Last modified February 27 08:19 EST 2024. Contains 370367 sequences. (Running on oeis4.)