

A064521


Number of rooted 5connected planar triangulations with 2n faces.


0



1, 0, 6, 13, 55, 189, 694, 2516, 9213, 33782, 124300, 458502, 1695469, 6284175, 23344173, 86904615, 324197100, 1211841846, 4538611107, 17029834923, 64014608376, 241046175666, 909171583214, 3434698413540, 12995770332449
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OFFSET

10,3


COMMENTS

No planar triangulation can be more than 5connected. The 5connected triangulations are historically important to the 4color problem.


LINKS

Table of n, a(n) for n=10..34.
Z. J. Gao, I. M. Wanless and N. C. Wormald, Counting 5connected planar triangulations, J. Graph Theory, Vol. 38 (2001), pp. 1835.


EXAMPLE

The smallest 5connected planar triangulation is the icosahedron, which has 20 faces. Because of its symmetry it has a unique rooting, so a(10)=1. The triangulations counted by a(12) and a(13) are drawn in the paper cited above.


MAPLE

# G.f. for 5connected planar triangulations: fiveconntri(m) returns the first m terms of a power series in w, in which the coefficient of w^n is the number of (rooted) 5connected planar triangulations with 2n faces.
fiveconntri := proc(howmanyterms) local keepterms, T, iteration, sval, previous; keepterms := howmanyterms+1; T := 3*w^3/(1+w)+ww^2+3*w^3w^4+4*(s+1)^3*((3*s1)*w+(3*s2)*(s+1)^3)*w/((3*s+2+ws^3)^3); iteration := s(w^2+2*(4*s^2+2*s+1)*(s+1)^2*w+s*(s+2)*(s+1)^4)/(8*w+2); sval := 0; previous := 1; while(sval<>previous) do previous := sval; sval := mtaylor(subs(s=sval, iteration), [w, s], keepterms); od: series(subs(s=sval, T), w, keepterms); end;


CROSSREFS

Sequence in context: A187985 A320043 A296619 * A262238 A111366 A177127
Adjacent sequences: A064518 A064519 A064520 * A064522 A064523 A064524


KEYWORD

nonn


AUTHOR

Ian Wanless, Oct 07 2001


STATUS

approved



