OFFSET
0,3
FORMULA
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Pell(k)*x^k*A(x)^k, where Pell = A000129.
G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x - x^2)/(1 - x - x^2)).
a(n) ~ sqrt((1 + 2^(1/3))*(4 + 7*2^(1/3))) * (2 + 3/2^(2/3) + 3/2^(1/3))^n / (3 * sqrt(Pi) * (2*n)^(3/2)). - Vaclav Kotesovec, Nov 05 2021
MATHEMATICA
terms = 25; A[_] = 0; Do[A[x_] = 1 + x A[x]/(1 - 2 x A[x] - x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 26; A[_] = 0; Do[A[x_] = 1 + Sum[Fibonacci[k, 2] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x - x^2)/(1 - x - x^2), {x, 0, terms}], x], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 07 2019
STATUS
approved