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A121112 Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition). 5
0, 5, 25, 155, 1080, 8085, 63525, 516790, 4315805, 36786385, 318736105, 2799049985, 24857641900, 222861398060, 2014418084860, 18337277269475, 167961106916065, 1546879330598945, 14315792338559005, 133065134882334095, 1241694764334690820, 11628016504072124555, 109243880617142972435 (list; graph; refs; listen; history; text; internal format)



From Petros Hadjicostas, Jul 29 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^* = a(r) (current sequence), U_r^{**} = A121113(r), and U_r^{***} = A121114(r) for r >= 1.

For the current sequence, we have a(r) = U_r^* = 5*U_{r-1} = 5*A036758(r-1) for r >= 2 with a(1) = U_1^* = 0.

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)).



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.


Table of n, a(n) for n=1..23.


a(r) = 5*A036758(r-1) for r >= 2 with a(1) = 0. - Petros Hadjicostas, Jul 29 2019


# Modification of N. J. A. Sloane's Maple program from A036758:

Order := 30: S := solve(series(G/(1+5*G+6*G^2+G^3), G)=x, G);

series(5*S*x, x = 0, 30) # Petros Hadjicostas, Jul 29 2019


Cf. A036758, A036760, A121113, A121114.

Sequence in context: A297589 A092166 A204209 * A090014 A249475 A179324

Adjacent sequences:  A121109 A121110 A121111 * A121113 A121114 A121115




N. J. A. Sloane, Aug 13, 2006


More terms from Petros Hadjicostas, Jul 29 2019 using N. J. A. Sloane's Maple program from A036758



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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)