A026106 Number of polyhexes of class PF2 (with one catafusene annealated to pyrene).

2, 5, 16, 55, 208, 817, 3336, 13935, 59406, 257079, 1126948, 4992421, 22318048, 100546543, 456055730, 2080872845, 9544572590, 43984730855, 203550840696, 945562887981, 4407586685688, 20609668887723, 96646196091276, 454402001079165

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn

Comments

See reference for precise definition.

From Petros Hadjicostas, Jan 12 2019: (Start)

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.)

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.

Sequences (N(m): m >= 1) and (M(m): m >= 1) appear in Table 1, p. 533.

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.

Sequence (1^Q_(4+n): n >= 1) appears in Table II, p. 537.

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.

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.

The second g.f. appears in eq. (A1), p. 1180, in Cyvin et al. (1994). It is algebraically equivalent to the first g.f.

(Apparently, the word "annealated" in Cyvin et al. (1992) is spelled "annelated" in Cyvin et al. (1994).)

(End)

Links

• S. J. Cyvin, Zhang Fuji, B. N. Cyvin, Guo Xiaofeng, and J. Brunvoll, Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
• S. J. Cyvin, B. N. Cyvin, J. Brunvoll, and E. Brendsdal, Enumeration and classification of certain polygonal systems representing polycyclic conjugated hydrocarbons: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
• Eric Weisstein's World of Mathematics, Fusenes.
• Eric Weisstein's World of Mathematics, Polyhex.

Crossrefs

Cf. A002212, A007317, A026106, A026118, A026298, A030519, A030520, A030525, A030529, A030532, A030534, A039658.

Programs

Maple

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;
taylor(bb(x), x = 0, 50); # Petros Hadjicostas, Jan 12 2019

Mathematica

(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 *)

Formula

From Petros Hadjicostas, Jan 12 2019: (Start)

For n >= 1, a(n+4) = (1/2)*(3*A002212(n) + A007317(floor((n+1)/2))).

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))).

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)).

(End)

Extensions

Name edited by Petros Hadjicostas, Jan 12 2019

Terms a(17)-a(28) computed by Petros Hadjicostas, Jan 12 2019