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)
D-finite with recurrence 16*n*(n+1)*(3779913*n-17115748)*a(n) +8*n*(22774701*n^2-189896835*n+300480484)*a(n-1) +8*(-84698862*n^3+724281753*n^2-1741575803*n+1262022932)*a(n-2) +4*(-247233813*n^3+2772092726*n^2-9119865299*n+8557786292)*a(n-3) +4*(-162946512*n^3-503762412*n^2+13063066949*n-33545271216)*a(n-4) +4*(-2069450478*n^3+29858261406*n^2-139077264791*n+207655047970)*a(n-5) +28*(498538263*n^3-8302056976*n^2+45726402090*n-83131156746)*a(n-6) +14*(-153328281*n^3+3031125480*n^2-20146667002*n+44899979512)*a(n-7) -98*(n-8)*(39339906*n^2-467514771*n+1280567548)*a(n-8) +49*(26062599*n-118057738)*(n-8)*(n-9)*a(n-9)=0. - R. J. Mathar, May 22 2025
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
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Feb 28 2004
STATUS
approved