A007894 Number of fullerenes with 2n vertices (or carbon atoms).

1, 0, 1, 1, 2, 3, 6, 6, 15, 17, 40, 45, 89, 116, 199, 271, 437, 580, 924, 1205, 1812, 2385, 3465, 4478, 6332, 8149, 11190, 14246, 19151, 24109, 31924, 39718, 51592, 63761, 81738, 99918, 126409, 153493, 191839, 231017, 285914, 341658, 419013

Offset: 10

Author

Boris Shraiman (boris(AT)physics.att.com), Gunnar Brinkmann and A. Dress (dress(AT)mathematik.uni-bielefeld.de)

Keywords

• nonn
• easy
• nice

Comments

Enantiomorphic pairs are regarded as the same here. Cf. A057210.

Contradictory results from the program "buckygen" from Brinkmann et al. (2012) and the program "fullgen" from Brinkmann and Dress (1997) led to the detection of a non-algorithmic error in fullgen. This bug has now been fixed and the results are in complete agreement. a(10)-a(190) were independently confirmed by buckygen and fullgen, while a(191)-a(200) were computed only by buckygen. - Jan Goedgebeur, Aug 08 2012

References

• A. T. Balaban, X. Liu, D. J. Klein, D. Babic, T. G. Schmalz, W. A. Seitz and M. Randic, "Graph invariants for fullerenes", J. Chem. Inf. Comput. Sci., vol. 35 (1995) 396-404.
• M. Deza, M. Dutour and P. W. Fowler, Zigzags, railroads and knots in fullerenes, J. Chem. Inf. Comput. Sci., 44 (2004), 1282-1293.
• J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
• P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Cambridge Univ. Press, 1995, see p. 32.
• P. W. Fowler, D. E. Manolopoulos and R. P. Ryan, "Isomerization of fullerenes", Carbon, 30 1235 1992.
• A. M. Livshits and Yu. E. Lozovik, Cut-and-unfold approach to Fullerene enumeration, J. Chem. Inf. Comput. Sci., 44 (2004), 1517-1520.
• Milicevic, A., and N. Trinajstic. "Combinatorial enumeration in chemistry." Chapter 8 in Chemical Modelling: Application and Theory, Vol. 4 (2006): 405-469.
• M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.

Links

• Jan Goedgebeur, Table of n, a(n) for n = 10..200.
• Gunnar Brinkmann, Jan Goedgebeur, and Brendan D. McKay, The Generation of Fullerenes, arXiv:1207.7010 [math.CO], 2012.
• Gunnar Brinkmann and Andreas Dress, fullgen.
• Gunnar Brinkmann and Andreas W. M. Dress, A constructive enumeration of fullerenes, Journal of Algorithms, Vol. 23, No. 2 (1997), 345-358.
• Gunnar Brinkmann, Jan Goedgebeur, and Brendan D. McKay, buckygen.
• CombOS - Combinatorial Object Server, generate fullerenes
• Philip Engel and Peter Smillie, The number of non-negative curvature triangulations of S^2, arXiv:1702.02614 [math.GT], 2017.
• Paul Gailiunas, Kagome from Deltahedra, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 337-344.
• Jan Goedgebeur and Brendan D. McKay, Fullerenes with distant pentagons, arXiv:1508.02878 [math.CO], (12-August-2015).
• House of Graphs, Fullerenes.
• Diaaeldin Taha, Wei Zhao, J. Maxwell Riestenberg, and Michael Strube, Normed Spaces for Graph Embedding, arXiv:2312.01502 [cs.LG], 2023. See p. 22.
• Eric Weisstein's World of Mathematics, Fullerene
• Wikipedia, Fullerene

Crossrefs

Cf. A057210, A046880.

Formula

a(n) = (809/2612138803200)*sigma_9(n) + O(n^8) where sigma_9(n) is the ninth divisor power sum, cf. A013957. - Philip Engel, Nov 29 2017

Extensions

Corrected a(68)-a(100) and added a(101)-a(200). - Jan Goedgebeur, Aug 08 2012