A203977 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.

1, 0, 6, 13, 49, 99, 304, 544, 1323, 2375, 4924, 8232, 15796, 25162, 44280, 68655, 113737, 169209, 269206, 389122, 592572, 837930, 1235648, 1702998, 2447847, 3311322, 4633866, 6167863, 8460396, 11064300, 14913540, 19247942, 25480686, 32492047, 42423238, 53411913, 68846728, 85840706, 109229064, 134916992, 169952280, 207903999, 259337312, 314901090, 388993995

Offset

12,3

Comments

Comments from Brendan McKay: (Start)

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.

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.

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)

References

B. D. McKay, Posting to Sequence Fans Mailing List, Oct 28 2011.

B. D. McKay, http://users.cecs.anu.edu.au/~bdm/rooted_maps_big.maple

Links

B. D. McKay and Jan Goedgebeur, Table of n, a(n) for n = 12..202 (terms n = 12..197 from B. D. McKay)

Gunnar Brinkmann, Jan Goedgebeur, Brendan D. McKay, The Generation of Fullerenes, arXiv:1207.7010

Crossrefs

Sequence in context: A239530 A131188 A247939 * A003757 A187985 A320043

Adjacent sequences: A203974 A203975 A203976 * A203978 A203979 A203980

Keyword

nonn

Author

N. J. A. Sloane, Jan 08 2012

Status

approved