OFFSET
1,5
COMMENTS
From Petros Hadjicostas, Jul 30 2019: (Start)
Quoting from p. 59 in Cyvin et al. (1997): "When an octagon is rooted at an edge ... then either (a) one branch can be attached in five directions at a time, (b) two branches can be attached in six ways, or (c) three branches in one way. Let the numbers of these kinds of systems be denoted by (a) U_r^*, (b) U_r^{**}, and (c) U_r^{***}, respectively."
Here r is "the number of octagons or eight-membered rings" in an edge-rooted catapolygon (here, catapolyoctagon). A catapolyoctagon is a "catacondensed polygonal system consisting of octagons" (where "catacondensed" means it has no internal vertices).
On p. 59 in Cyvin et al. (1997), the total number of edge-rooted catapolyoctagons (each with r octagons) is denoted by U_r, and we have U_r = U_r^* + U_r^{**} + U_r^{***} for r >= 2.
We have U_r = A036758(r), U_r^* = A121112(r), U_r^{**} = A121113(r), and U_r^{***} = a(r) (current sequence) for r >= 1.
For the current sequence, we have a(r) = U_r^{***} = Sum_{i = 1..r-3} U(i) * Sum_{j = 1..r-i-2} U(j) * U(r-1-i-j) for r >= 4, where U(r) = A036758(r), with a(1) = a(2) = a(3) = 0. See Eq. (13) on p. 59 in Cyvin et al. (1997).
The ultimate purpose of these calculations (in the paper by Cyvin et al. (1997)) is the calculation of I_r = A036760(r), which is the "number of nonisomorphic free (unrooted) catapolyoctagons when r is given." These catapolyoctagons "represent a class of polycyclic conjugated hydrocarbons, C_{6r+2} H_{4r+4}" (see p. 57 in Cyvin et al. (1997)).
The g.f.'s of the sequences U, U^*, U^{**}, and U^{***} appear also in Eqs. (2) and (3) on p. 194 in Brunvoll et al. (1997).
(End)
REFERENCES
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70; see pp. 59-61.
LINKS
J. Brunvoll, S. J. Cyvin, and B. N. Cyvin, Enumeration of tree-like octagonal systems, J. Math. Chem., 21 (1997), 193-196; see Eqs. (2) and (3) on p. 194.
FORMULA
a(r) = Sum_{i = 1..r-3} U(i) * Sum_{j = 1..r-i-2} U(j) * U(r-1-i-j) for r >= 4, where U(r) = A036758(r), with a(1) = a(2) = a(3) = 0. - Petros Hadjicostas, Jul 30 2019
MAPLE
# Modification of N. J. A. Sloane's Maple program from A036758:
Order := 30;
S := solve(series(G/(G^3 + 6*G^2 + 5*G + 1), G) = x, G);
series(S^3*x, x = 0, 30); # Petros Hadjicostas, Jul 30 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 13 2006
EXTENSIONS
More terms from Petros Hadjicostas, Jul 30 2019
STATUS
approved