cp's OEIS Frontend

See reference for precise definition.

From Petros Hadjicostas, Jan 13 2019: (Start)

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).

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).

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.)

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).

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.

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).

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

(End)

See reference for precise definition.

Cyvin has incorrect a(13)=273 and a(14)=608 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II. - Sean A. Irvine, Apr 03 2020

See reference for precise definition.

Cyvin has incorrect a(13)=369366 and a(14)=1709123 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II.

See reference for precise definition.

Cyvin has incorrect a(13)=369639 and a(14)=1710137 in Table III due to using incorrect values for A026298(13) and A026298(14) in Table II. - Sean A. Irvine, Apr 02 2020

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)

See reference for precise definition.

Cyvin et al. has incorrect a(13) = 119204 and a(14) = 575312 due to using incorrect value for A039919(5); cf. A039659. - Sean A. Irvine, Sep 24 2019

See reference for precise definition.

See reference for precise definition.

Column C_s of Table 6 in Cyvin et al. Computed by subtracting all the other symmetry classes from the total. - Sean A. Irvine, Mar 27 2021