OFFSET
0,3
COMMENTS
The ratio of the ratios of the consecutive numbers of this sequence tends to the silver ratio (A014176 = 1 + sqrt(2)). This property can be proven.
FORMULA
a(n) = A000129(n)*a(n-1) + a(n-2) for n > 1.
a(n)*a(n-2)/a(n-1)^2 ~ 1 + sqrt(2).
a(n) ~ c * (1 + sqrt(2))^(n*(n+1)/2) / 2^(3*n/2), where c = 1.2795822677496757181586660872019163393334688780614201258263927413618707127... - Vaclav Kotesovec, Dec 29 2019
MATHEMATICA
Nest[Append[#1, Fibonacci[#2, 2] #1[[-1]] + #1[[-2]] ] & @@ {#, Length@ #} &, {0, 1}, 14] (* Michael De Vlieger, Dec 22 2019 *)
PROG
(PARI) seq(n)={my(u=Vec(1/(1 - 2*x - x^2) + O(x^n)), v=vector(#u+1)); v[2]=1; for(n=2, #u, v[n+1] = u[n]*v[n] + v[n-1]); v} \\ Andrew Howroyd, Dec 22 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Archit Handa, Dec 22 2019
STATUS
approved