A003123
Number of Hamiltonian rooted triangulations with n internal nodes and 4 external nodes.
Original entry on oeis.org
2, 12, 92, 800, 7554, 75664, 792448, 8595120, 95895816, 1095130728, 12753454896, 151017596448, 1814135701956, 22067487234504, 271407264938656, 3370796862212944, 42230992336570032, 533252038221313888, 6781213722509638192, 86790636905453265216
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
P(n,k) = k*(2*n+2*k-4)!*(2*n+k-1)!/((n+k-1)!*(n+k-2)!*n!*(n+k)!);
F(K, N=23) = {
my(x='x + O('x^(K+1)), t='t + O('t^(N+1)),
r='t*Ser(vector(N, n, sqr(binomial(2*n,n)/(n+1))),'t),
p=x^3*Ser(apply(k->Ser(vector(N, n, P(n-1,k)),'t), [3..K])),
s=serreverse(t*(1+r)), f=subst(subst(p, 't, s), 'x, 'x*s/'t));
Vec(polcoeff(f,K));
};
F(4) \\ Gheorghe Coserea, Aug 18 2017
A005979
Number of Hamiltonian rooted triangulations with n internal nodes and 5 external nodes.
Original entry on oeis.org
5, 45, 420, 4130, 42480, 453350, 4986860, 56251230, 648055650, 7601584050, 90556803600, 1093417607850, 13359234113250, 164935358510470, 2055350730457020, 25827268868350690, 326989527294142480, 4168165020885948440, 53462092704596804720
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A028475
Total number of Hamiltonian cycles avoiding the root-edge in rooted cubic bipartite planar maps with 2n nodes.
Original entry on oeis.org
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
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.
- 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.
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