

A121114


Edgerooted treelike octagonal systems (see the Cyvin et al. reference for precise definition).


5



0, 0, 0, 1, 15, 168, 1703, 16539, 157416, 1483900, 13928238, 130547475, 1223803350, 11484513612, 107940809223, 1016351200410, 9588249961074, 90633332095992, 858386837556696, 8145257860480545, 77432954101974513, 737419153249761752, 7034562802431438771, 67214038308803342715
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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 eightmembered rings" in an edgerooted 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 edgerooted 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..r3} U(i) * Sum_{j = 1..ri2} U(j) * U(r1ij) 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 treelike octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 5570; see pp. 5961.


LINKS

Table of n, a(n) for n=1..24.
J. Brunvoll, S. J. Cyvin, and B. N. Cyvin, Enumeration of treelike octagonal systems, J. Math. Chem., 21 (1997), 193196; see Eqs. (2) and (3) on p. 194.


FORMULA

a(r) = Sum_{i = 1..r3} U(i) * Sum_{j = 1..ri2} U(j) * U(r1ij) 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

Cf. A036758, A036759, A036760, A121112, A121113.
Sequence in context: A016234 A160197 A055660 * A121116 A218928 A202663
Adjacent sequences: A121111 A121112 A121113 * A121115 A121116 A121117


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Aug 13 2006


EXTENSIONS

More terms from Petros Hadjicostas, Jul 30 2019


STATUS

approved



