%I #24 Jul 31 2019 03:52:46
%S 1,1,3,4,15,23,94,155,661,1139,4983,8844,39362,71360,321561,592361,
%T 2694421,5025849,23029195,43388208,199990961,379900479,1759636142,
%U 3365582261,15652514944,30112397278,140531706444,271707661708
%N Number of mirror-symmetrical edge-rooted tree-like octagonal systems.
%D S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134(1) (1997), 55-70. [The index of summation in Eq. (15), p. 60, should start at i = 0, not at i = 1. - _Petros Hadjicostas_, Jul 30 2019]
%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.
%F G.f. V=V(x) satisfies x(x-2)V^3 + 2(x^2-3x+1)V^2 + (-x^2-3x+2)V - x(x+2) = 0.
%F From _Petros Hadjicostas_, Jul 30 2019: (Start)
%F Let U(0) = 1 and U(n) = A036758(n) for n >= 1. Let also a(0) = a(1) = 1 (even though the offset for the current sequence is 1 as it is done in Table II (p. 61) in Cyvin et al. (1997) and in Eq. (5), p. 195, in Brunvoll et al. (1997)).
%F Then
%F a(n) = Sum_{i = 0..floor((n-1)/2)} U(i) * a(n-1-2*i) for n even >= 2, and
%F a(n) = U((n-1)/2) + Sum_{i = 0..floor((n-1)/2)} U(i) * a(n-1-2*i) for n odd >= 3.
%F This is Eq. (15), p. 60, in Cyvin et al. (1997), but we have corrected the lower index of summation (from i = 1 to i = 0).
%F (End)
%p F := (2+3*V+6*V^2+2*V^3-(V+2)*sqrt(1+4*V+8*V^2+4*V^4))/2/(V^3+2*V^2-V-1): Order := 40: S := solve(series(F,V)=x,V);
%o (PARI) a(n)=if(n<1,0,polcoeff(serreverse((2*x^3+6*x^2+3*x+2-(x+2)*sqrt(4*x^4+8*x^2+4*x+1+x*O(x^n)))/2/(x^3+2*x^2-x-1)),n)) /* _Michael Somos_, Mar 10 2004 */
%Y Cf. A036758, A036760, A121112, A121113, A121114.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Feb 28 2004