|
%I #64 Apr 24 2020 07:52:38
%S 2,5,16,55,208,817,3336,13935,59406,257079,1126948,4992421,22318048,
%T 100546543,456055730,2080872845,9544572590,43984730855,203550840696,
%U 945562887981,4407586685688,20609668887723,96646196091276,454402001079165
%N Number of polyhexes of class PF2 (with one catafusene annealated to pyrene).
%C See reference for precise definition.
%C From _Petros Hadjicostas_, Jan 12 2019: (Start)
%C In Cyvin et al. (1992), sequence (N(m): m >= 1) = (A002212(m): m >= 1) is defined by eq. (1), p. 533. (We may let N(0) := A002212(0) = 1.)
%C Sequence (M(m): m >= 1) is defined by eq. (13), p. 534. We have M(2*m) = M(2*m-1) = A007317(m) for m >= 1.
%C Sequences (N(m): m >= 1) and (M(m): m >= 1) appear in Table 1, p. 533.
%C The current sequence is denoted by 1^Q_(4+n) (with n = 1,2,3,...). Thus, a(n+4) = 1^Q_(4+n) for n >= 1; i.e., a(m) = 1^Q_{m} for m >= 5. We have 1^Q_(4+n) = (1/2)*(3*N(n) + M(n)) for n >= 1. See eq. (33), p. 536.
%C Sequence (1^Q_(4+n): n >= 1) appears in Table II, p. 537.
%C We may use the many formulae in the documentations of sequences A002212 and A007317 in order to create complicated formulae and recurrence relations for (a(n): n >= 5). We omit the details.
%C The first g.f. below is a combination of the g.f. for sequence A002212 by John W. Layman in 2001 and the g.f. for sequence A007317 by Ira M. Gessel and Jang Soo Kim in 2010.
%C The second g.f. appears in eq. (A1), p. 1180, in Cyvin et al. (1994). It is algebraically equivalent to the first g.f.
%C (Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)
%C (End)
%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fusene.html">Fusenes</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polyhex.html">Polyhex</a>.
%F From _Petros Hadjicostas_, Jan 12 2019: (Start)
%F For n >= 1, a(n+4) = (1/2)*(3*A002212(n) + A007317(floor((n+1)/2))).
%F G.f.: (x^3/4)*(4 - 8*x - 3*sqrt(1 - 6*x + 5*x^2) - (x + 1)*sqrt((1 - 5*x^2)/(1 - x^2))).
%F G.f.: x^3*(1 - 2*x) - (x^3/4)*(3*(1 - x)^(1/2)*(1 - 5*x)^(1/2) + (1 - x)^(-1)*(1 - x^2)^(1/2)*(1 - 5*x^2)^(1/2)) (see eq. (A1), p. 1180, in Cyvin et al. (1994)).
%F (End)
%p bb := proc(x) (1/4)*x^3*(4-8*x-3*sqrt((1-x)*(1-5*x))-(x+1)*sqrt((1-5*x^2)/(1-x^2))) end proc;
%p taylor(bb(x), x = 0, 50); # _Petros Hadjicostas_, Jan 12 2019
%t (1/4) x^3 (4 - 8x - 3Sqrt[(1-x)(1-5x)] - (x+1) Sqrt[(1-5x^2)/(1-x^2)]) + O[x]^29 // CoefficientList[#, x]& // Drop[#, 5]& (* _Jean-François Alcover_, Apr 24 2020, from Maple *)
%Y Cf. A002212, A007317, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.
%K nonn
%O 5,1
%A _N. J. A. Sloane_
%E Name edited by _Petros Hadjicostas_, Jan 12 2019
%E Terms a(17)-a(28) computed by _Petros Hadjicostas_, Jan 12 2019
| 