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%I #33 Jul 30 2019 21:08:53
%S 0,5,25,155,1080,8085,63525,516790,4315805,36786385,318736105,
%T 2799049985,24857641900,222861398060,2014418084860,18337277269475,
%U 167961106916065,1546879330598945,14315792338559005,133065134882334095,1241694764334690820,11628016504072124555,109243880617142972435
%N Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition).
%C From _Petros Hadjicostas_, Jul 29 2019: (Start)
%C 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."
%C 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).
%C 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.
%C We have U_r = A036758(r), U_r^* = a(r) (current sequence), U_r^{**} = A121113(r), and U_r^{***} = A121114(r) for r >= 1.
%C 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.
%C 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)).
%C (End)
%D 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.
%F a(r) = 5*A036758(r-1) for r >= 2 with a(1) = 0. - _Petros Hadjicostas_, Jul 29 2019
%p # Modification of _N. J. A. Sloane_'s Maple program from A036758:
%p Order := 30: S := solve(series(G/(1+5*G+6*G^2+G^3), G)=x, G);
%p series(5*S*x, x = 0, 30) # _Petros Hadjicostas_, Jul 29 2019
%Y Cf. A036758, A036760, A121113, A121114.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Aug 13, 2006
%E More terms from _Petros Hadjicostas_, Jul 29 2019 using _N. J. A. Sloane_'s Maple program from A036758
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