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 A036759 Number of mirror-symmetrical edge-rooted tree-like octagonal systems. 8
 1, 1, 3, 4, 15, 23, 94, 155, 661, 1139, 4983, 8844, 39362, 71360, 321561, 592361, 2694421, 5025849, 23029195, 43388208, 199990961, 379900479, 1759636142, 3365582261, 15652514944, 30112397278, 140531706444, 271707661708 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 REFERENCES 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] LINKS J. Brunvoll, S. J. Cyvin, and B. N. Cyvin, Enumeration of tree-like octagonal systems, J. Math. Chem., 21 (1997), 193-196. FORMULA 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. From Petros Hadjicostas, Jul 30 2019: (Start) 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)). Then a(n) = Sum_{i = 0..floor((n-1)/2)} U(i) * a(n-1-2*i) for n even >= 2, and a(n) = U((n-1)/2) + Sum_{i = 0..floor((n-1)/2)} U(i) * a(n-1-2*i) for n odd >= 3. 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). (End) MAPLE 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); PROG (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 */ CROSSREFS Cf. A036758, A036760, A121112, A121113, A121114. Sequence in context: A109926 A272514 A065942 * A263718 A286675 A286025 Adjacent sequences:  A036756 A036757 A036758 * A036760 A036761 A036762 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Emeric Deutsch, Feb 28 2004 STATUS approved

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Last modified June 18 01:16 EDT 2021. Contains 345098 sequences. (Running on oeis4.)