OFFSET
0,8
COMMENTS
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (1 - x/(1 - 1/B(x))))/x^3 where B(x) equals the g.f. of the row sums of triangle A186084.
G.f.: 1/(1-x^2 - x^5/(1-x^3 - x^7/(1-x^4 - x^9/(1-x^5 - x^11/(1-x^6 - x^13/(1-...)))))) (continued fraction).
G.f.: 1/(1-x^2/(1-x^3/(1-x^7/(1-x^4/(1-x^5/(1-x^11/(1-x^6/(1 -x^7/(1-x^15/(1-...)))))))))) (continued fraction).
G.f.: 1/x^3 - (Q(0) + 1)/x^2, where Q(k)= 1/x^(k+1) - 1 - 1/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 07 2013
a(n) ~ c * d^n, where d = 1.3712018040437285..., c = 0.154355235026898... . - Vaclav Kotesovec, Sep 10 2014
EXAMPLE
G.f.: 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 +...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),
`if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))
end:
a:= n-> add(coeff(b(n-k, 1), x, k), k=0..n):
seq(a(n), n=0..70); # Alois P. Heinz, Jul 24 2013
MATHEMATICA
m = 100;
f[i_] := If[i == 0, 1, -x^(2i+3)];
g[i_] := 1 - x^(i+2);
ContinuedFractionK[f[i], g[i], {i, 0, Sqrt[m] // Ceiling}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Oct 14 2019, after Sergei N. Gladkovskii *)
PROG
(PARI) {a(n)=local(Txy=1+x*y); for(i=1, n, Txy=1/(1-x*y-x^3*y^2*subst(Txy, y, x*y+x*O(x^n)))); polcoeff(subst(Txy, y, x), n, x)}
(PARI) N = 66; x = 'x + O('x^N);
Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) );
gf = 1/x^3 - (Q(0) + 1)/x^2;
Vec(gf) \\ Joerg Arndt, May 07 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2011
STATUS
approved