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

%I A002216 M1426 N0562 #55 Feb 16 2025 08:32:25
%S A002216 0,1,1,2,5,12,37,123,446,1689,6693,27034,111630,467262,1981353,
%T A002216 8487400,36695369,159918120,701957539,3101072051,13779935438,
%U A002216 61557789660,276327463180,1245935891922,5640868033058,25635351908072,116911035023017
%N A002216 Harary-Read numbers: restricted hexagonal polyominoes (cata-polyhexes) with n cells.
%C A002216 Named after the American mathematician Frank Harary (1921-2005) and the British mathematician Ronald Cedric Read (1924-2019). - _Amiram Eldar_, Jun 22 2021
%D A002216 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.
%D A002216 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.
%D A002216 J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
%D A002216 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.
%D A002216 J. V. Knop, K. Szymansky, Željko Jeričević and Nenad Trinajstić, On the total number of polyhexes, Match, Vol. 16 (1984), pp. 119-134.
%D A002216 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002216 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A002216 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.
%H A002216 T. D. Noe, <a href="/A002216/b002216.txt">Table of n, a(n) for n = 0..200</a>
%H A002216 L. W. Beineke and R. E. Pippert, <a href="https://doi.org/10.1017/S0017089500002305">On the enumeration of planar trees of hexagons</a>, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147.
%H A002216 L. W. Beineke and R. E. Pippert, <a href="/A004127/a004127.pdf">On the enumeration of planar trees of hexagons</a>, Glasgow Math. J., Vol. 15, No. 2 (1974), pp. 131-147. [Annotated scanned copy]
%H A002216 S. J. Cyvin, J. Brunvoll and B. N. Cyvin, <a href="https://doi.org/10.1007/BF01172927">Harary-Read numbers for catafusenes: Complete classification according to symmetry</a>, Journal of mathematical chemistry, Vol. 9, No. 1 (1992), pp. 19-31 and 33-38. See Table 2.
%H A002216 F. Harary and R. C. Read, <a href="https://doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinb. Math. Soc., Vol. 17, No. 1 (1970), pp. 1-13; <a href="https://www.researchgate.net/publication/238343540_The_enumeration_of_tree-like_polyhexes">alternative link</a>.
%H A002216 J. V. Knop, K. Szymanski, Ž. Jeričević, and N. Trinajstić, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match16/match16_119-134.pdf">On the total number of polyhexes</a>, Match, No. 16 (1984), 119-134.
%H A002216 R. C. Read, <a href="/A002216/a002216.pdf">Letter to N. J. A. Sloane, Feb 12 1971</a>. (includes 40 terms of A002212 and A002216)
%H A002216 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.
%H A002216 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fusene.html">Fusene</a>.
%F A002216 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)).
%F A002216 a(n) = (1/2)[A002214(n)+A002215(n)], n>=1. - _Emeric Deutsch_, Dec 23 2003
%F A002216 a(n) ~ 5^(n+1/2)/(4*sqrt(Pi)*n^(5/2)). - _Vaclav Kotesovec_, Aug 09 2013
%t A002216 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 *)
%Y A002216 Cf. A036359, A005963, A000228, A001998.
%Y A002216 Cf. A002212, A002213, A002214, A002215.
%K A002216 nonn,easy,nice
%O A002216 0,4
%A A002216 _N. J. A. Sloane_