A036759 Number of mirror-symmetrical edge-rooted tree-like octagonal systems.

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

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

Table of n, a(n) for n=1..28.

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: A272514 A065942 A369082 * A263718 A286675 A286025

Adjacent sequences: A036756 A036757 A036758 * A036760 A036761 A036762

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Feb 28 2004

Status

approved