A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.

0, 1, 1, 2, 5, 12, 37, 123, 446, 1689, 6693, 27034, 111630, 467262, 1981353, 8487400, 36695369, 159918120, 701957539, 3101072051, 13779935438, 61557789660, 276327463180, 1245935891922, 5640868033058, 25635351908072, 116911035023017

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Named after the American mathematician Frank Harary (1921-2005) and the British mathematician Ronald Cedric Read (1924-2019). - Amiram Eldar, Jun 22 2021

References

• S. J. Cyvin, J. Brunvoll, X. F. Guo and F. J. Zhang, Number of perifusenes with one internal vertex, Rev. Roumaine Chem., Vol. 38, No. 1 (1993), pp. 65-77.
• S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem., Vol. 134, No. 1 (1997), pp. 55-70.
• J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
• Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron, Vol. 42, No. 19 (1986), pp. 5291-5299. See Table 3.
• J. V. Knop, K. Szymansky, Željko Jeričević and Nenad Trinajstić, On the total number of polyhexes, Match, Vol. 16 (1984), pp. 119-134.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer generation of isomeric structures, Pure & Appl. Chem., Vol. 55, No. 2 (1983), pp. 379-390.

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147.
• L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147. [Annotated scanned copy]
• S. J. Cyvin, J. Brunvoll and B. N. Cyvin, Harary-Read numbers for catafusenes: Complete classification according to symmetry, Journal of mathematical chemistry, Vol. 9, No. 1 (1992), pp. 19-31 and 33-38. See Table 2.
• F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc., Vol. 17, No. 1 (1970), pp. 1-13; alternative link.
• J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, On the total number of polyhexes, Match, No. 16 (1984), 119-134.
• R. C. Read, Letter to N. J. A. Sloane, Feb 12 1971. (includes 40 terms of A002212 and A002216)
• Eric Weisstein's World of Mathematics, Polyhex.
• Eric Weisstein's World of Mathematics, Fusene.

Crossrefs

Cf. A036359, A005963, A000228, A001998.

Cf. A002212, A002213, A002214, A002215.

Programs

Mathematica

CoefficientList[Series[(12+(1-5*x)^(3/2)*(1-x)^(3/2)+24*x-48*x^2- 24*x^3- 3*(3+5 x)*Sqrt[1-5*x^2]*Sqrt[1-x^2]-4*Sqrt[1-5*x^3]*Sqrt[1-x^3])/ (24*x^2),{x,0,40}],x] (* Harvey P. Dale, Dec 23 2013 *)

Formula

G.f.: (1/(24*x^2))*(12+24*x-48*x^2-24*x^3 +(1-x)^(3/2)*(1-5*x)^(3/2)-3*(3+5*x)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2) -4*(1-x^3)^(1/2)*(1-5*x^3)^(1/2)).

a(n) = (1/2)[A002214(n)+A002215(n)], n>=1. - Emeric Deutsch, Dec 23 2003

a(n) ~ 5^(n+1/2)/(4*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 09 2013