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A064521 Number of rooted 5-connected 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 (list; graph; refs; listen; history; text; internal format)



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


Z. J. Gao, I. M. Wanless and N. C. Wormald, Counting 5-connected planar triangulations, J. Graph Theory, Vol. 38 (2001), pp. 18-35.


Table of n, a(n) for n=10..34.


The smallest 5-connected 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.


# G.f. for 5-connected 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) 5-connected planar triangulations with 2n faces.

fiveconntri := proc(howmanyterms) local keepterms, T, iteration, sval, previous; keepterms := howmanyterms+1; T := -3*w^3/(1+w)+w-w^2+3*w^3-w^4+4*(s+1)^3*((3*s-1)*w+(3*s-2)*(s+1)^3)*w/((3*s+2+w-s^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;


Sequence in context: A203977 A003757 A187985 * A262238 A111366 A177127

Adjacent sequences:  A064518 A064519 A064520 * A064522 A064523 A064524




Ian M. Wanless (wanless(AT)maths.ox.ac.uk), Oct 07 2001



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Last modified November 30 21:08 EST 2015. Contains 264673 sequences.