login
Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).
10

%I #50 Sep 17 2019 19:12:11

%S 5,20,100,431,1937,8548,38199,171001,770934,3492251,15905897,72785480,

%T 334571647,1544203452,7154247842,33260560977,155126129968,

%U 725639264293,3403612632885,16004969728270,75437244856898,356337397010035,1686618801843050

%N Number of polyhexes of class PF2 (with two catafusenes annealated to pyrene).

%C See reference for precise definition.

%C From _Petros Hadjicostas_, Jan 13 2019: (Start)

%C 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 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 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 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 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 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 (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 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 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 14 2019: (Start)

%F 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 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 (End)

%Y Cf. A002212, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.

%K nonn

%O 6,1

%A _N. J. A. Sloane_

%E Name edited by _Petros Hadjicostas_, Jan 13 2019

%E Terms a(17)-a(28) computed by _Petros Hadjicostas_, Jan 13 2019 using a g.f. in Cyvin et al. (1994)