|
| |
|
|
A121113
|
|
Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition).
|
|
5
|
|
|
|
0, 0, 6, 60, 522, 4452, 38130, 329832, 2884056, 25476936, 227145654, 2041930920, 18490834362, 168537705300, 1545096559812, 14238592913328, 131826509242650, 1225645805016864, 11438847800351118, 107128560124123524, 1006475582377759578, 9483340466106708180, 89593844489912910294
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
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^{**} = a(r) (current sequence), and U_r^{***} = A121114(r) for r >= 1.
For the current sequence, we have a(r) = U_r^{**} = 6*Sum_{i = 1.. r-2} U(i) * U(r-1-i) for r >= 3, where U(r) = A036758(r), with a(1) = a(2) = 0. See Eq. (12) 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
|
|
|
|
FORMULA
|
a(r) = 6*Sum_{i = 1.. r-2} U(i) * U(r-1-i) for r >= 3, where U(r) = A036758(r), with a(1) = a(2) = 0. - Petros Hadjicostas, Jul 30 2019
|
|
|
MAPLE
|
Order := 30; S := solve(series(G/(G^3 + 6*G^2 + 5*G + 1), G) = x, G);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
