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%I #34 Jul 31 2019 03:53:03
%S 0,0,6,60,522,4452,38130,329832,2884056,25476936,227145654,2041930920,
%T 18490834362,168537705300,1545096559812,14238592913328,
%U 131826509242650,1225645805016864,11438847800351118,107128560124123524,1006475582377759578,9483340466106708180,89593844489912910294
%N Edge-rooted tree-like octagonal systems (see the Cyvin et al. reference for precise definition).
%C From _Petros Hadjicostas_, Jul 30 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^* = A121112(r), U_r^{**} = a(r) (current sequence), and U_r^{***} = A121114(r) for r >= 1.
%C 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).
%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 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).
%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.
%H J. Brunvoll, S. J. Cyvin, and B. N. Cyvin, <a href="http://dx.doi.org/10.1023/A:1019122419384">Enumeration of tree-like octagonal systems</a>, J. Math. Chem., 21 (1997), 193-196; see Eqs. (2) and (3) on p. 194.
%F 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
%p # Modification of _N. J. A. Sloane_'s Maple program from A036758:
%p Order := 30; S := solve(series(G/(G^3 + 6*G^2 + 5*G + 1), G) = x, G);
%p series(6*S^2*x, x = 0, 30); # _Petros Hadjicostas_, Jul 30 2019
%Y Cf. A036758, A036759, A036760, A121112, A121114.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Aug 13 2006
%E More terms from _Petros Hadjicostas_, Jul 30 2019
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