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%I #40 Aug 23 2020 20:48:08
%S 1,0,6,13,49,99,304,544,1323,2375,4924,8232,15796,25162,44280,68655,
%T 113737,169209,269206,389122,592572,837930,1235648,1702998,2447847,
%U 3311322,4633866,6167863,8460396,11064300,14913540,19247942,25480686,32492047,42423238,53411913,68846728,85840706,109229064,134916992,169952280,207903999,259337312,314901090,388993995
%N Number of rooted fullerenes with n faces, where "rooted" means that one triple (v,e,f) is distinguished, where v is a vertex, e is an edge on that vertex and f is a face on that edge.
%C Comments from _Brendan McKay_: (Start)
%C A fullerene is a cubic planar graph with only faces of size 5 and 6. It is also called a buckyball, and a standard soccer ball is an example.
%C A deep theoretical result is that c(n) is proportional to n^10 for very large n. Polynomial growth is very rare for graph classes. It is plausible that in fact c(n) is given by a formula. For example, it might be a polynomial of degree 10 (but it isn't, I checked). Or it might be a different polynomial of degree 10 according to n mod 4. (There is a distinct wriggle of period 4, but this one doesn't seem to work either; don't trust me.) Or some other possibility. My fantasy is that just by playing with the numbers in different ways it might be possible to guess the formula, if there is one.
%C Another possibility is that the generating function might be guessable. For example it might be a rational function c(x) = p(x)/q(x) where p,q are polynomials and the smallest zeros of q(x) have absolute value 1 and one such zero has multiplicity 11. (End)
%D B. D. McKay, Posting to Sequence Fans Mailing List, Oct 28 2011.
%D B. D. McKay, http://users.cecs.anu.edu.au/~bdm/rooted_maps_big.maple
%H B. D. McKay and Jan Goedgebeur, <a href="/A203977/b203977.txt">Table of n, a(n) for n = 12..202</a> (terms n = 12..197 from B. D. McKay)
%H Gunnar Brinkmann, Jan Goedgebeur, Brendan D. McKay, <a href="http://arxiv.org/abs/1207.7010">The Generation of Fullerenes</a>, arXiv:1207.7010
%K nonn
%O 12,3
%A _N. J. A. Sloane_, Jan 08 2012