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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)
OFFSET

1,2

COMMENTS

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)).

(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

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

FORMULA

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

MAPLE

# 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

CROSSREFS

Cf. A036758, A036760, A121113, A121114.

Sequence in context: A297589 A092166 A204209 * A090014 A249475 A179324

Adjacent sequences:  A121109 A121110 A121111 * A121113 A121114 A121115

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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