This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A026118 #51 Feb 16 2025 08:32:35
%S A026118 5,20,100,431,1937,8548,38199,171001,770934,3492251,15905897,72785480,
%T A026118 334571647,1544203452,7154247842,33260560977,155126129968,
%U A026118 725639264293,3403612632885,16004969728270,75437244856898,356337397010035,1686618801843050
%N A026118 Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).
%C A026118 See reference for precise definition.
%C A026118 From _Petros Hadjicostas_, Jan 13 2019: (Start)
%C A026118 This sequence is defined by eq. (34), p. 536, in Cyvin et al. (1992). It is denoted by 2^Q_{4+n} (for n >= 2). Thus, a(n+4) = 2^Q_{4+n} for n >= 2 (and that is why the offset here is 6).
%C A026118 For n >= 2, we have a(n+4) = (3/4)*(1 + (-1)^n)*N(floor(n/2)) + (1/4)*(L(n) + 13*Sum_{1 <= i <= n-1} N(i)*N(n-i)), where N(n) = A002212(n) and L(n) = A039658(n).
%C A026118 The sequence (N(n): n >= 1) = (A002212(n): n >= 1) is given by eq. (1), p. 533, in Cyvin et al. (1992), while its g.f. is given by eqs. (2)-(4), p. 1174, in Cyvin et al. (1994). (The g.f. of N(n) = A002212(n) appears also in Harary and Read (1970) as eq. (9) on p. 4.)
%C A026118 The sequence (L(n): n >= 1) = (A039658(n): n >= 1) is given by eq. (22), p. 535, in Cyvin et al (1992), while its g.f. is given by eq. (9), p. 1175, in Cyvin et al. (1994).
%C A026118 The g.f. of the current sequence (a(m): m >= 6) (see below) is given in eq. (A2), p. 1180, in Cyvin et al. (1994), but it can be derived by the above formulae using standard techniques for the calculation of g.f.'s.
%C A026118 For the number of polyhexes of class PF2, we have 1^Q_h = A026106(h) (h >= 5, one catafusene annealated to pyrene), 3^Q_h = A026298(h) (h >= 7, three catafusenes annealated to pyrene), and 4^Q_h = A030519(h) (h >= 8, four catafusenes annealated to pyrene).
%C A026118 (Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)
%C A026118 (End)
%H A026118 S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, <a href="https://pubs.acs.org/doi/pdfplus/10.1021/ci00009a021">Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices</a>, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
%H A026118 S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, <a href="https://pubs.acs.org/doi/pdf/10.1021/ci00021a026">Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes</a>, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
%H A026118 F. Harary and R. C. Read, <a href="http://dx.doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
%H A026118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fusene.html">Fusenes</a>.
%H A026118 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.
%F A026118 From _Petros Hadjicostas_, Jan 14 2019: (Start)
%F A026118 a(n+4) = (3/4)*(1 + (-1)^n)*N(floor(n/2)) + (1/4)*(L(n) + 13*Sum_{1 <= i <= n-1} N(i)*N(n-i)) for n >= 2, where N(n) = A002212(n) and L(n) = A039658(n).
%F A026118 G.f.: (x^2/4)*(1-x)^(-1)*(10 - 48*x + 74*x^2 - 38*x^3) - (x^2/8)*[13*(1 - 3*x)*(1 - x)^(1/2)*(1 - 5*x)^(1/2) + (1 - x)^(-1)*(7 - 5*x)*(1 - x^2)^(1/2)*(1 - 5*x^2)^(1/2)] (see eq. (A2), p. 1180, in Cyvin et al. (1994)).
%F A026118 (End)
%Y A026118 Cf. A002212, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.
%K A026118 nonn
%O A026118 6,1
%A A026118 _N. J. A. Sloane_
%E A026118 Name edited by _Petros Hadjicostas_, Jan 13 2019
%E A026118 Terms a(17)-a(28) computed by _Petros Hadjicostas_, Jan 13 2019 using a g.f. in Cyvin et al. (1994)