A130235 Partial sums of the 'lower' Fibonacci Inverse A130233.

0, 2, 5, 9, 13, 18, 23, 28, 34, 40, 46, 52, 58, 65, 72, 79, 86, 93, 100, 107, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 227, 236, 245, 254, 263, 272, 281, 290, 299, 308, 317, 326, 335, 344, 353, 362, 371, 380, 389, 398, 407, 417, 427

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Formula

a(n) = Sum_{k=0..n} A130233(k) = (n+1)*A130233(n) - Fib(A130233(n)+2) + 1.

G.f.: 1/(1-x)^2 * Sum_{k>=1} x^Fib(k). [corrected by Joerg Arndt, Apr 14 2020]

Mathematica

nmax = 90; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 1, 1 + Log[3/2 + Sqrt[5]*nmax]/Log[GoldenRatio]}]/(1-x)^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 14 2020 *)

Prog

(Magma)
m:=120;
f:= func< x | (&+[x^Fibonacci(j): j in [1..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
R<x>:=PowerSeriesRing(Rationals(), m+1);
[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 17 2023
(SageMath)
m=120
def f(x): return sum( x^fibonacci(j) for j in range(1, int(3*log(3*m+1))))/(1-x)^2
def A130235_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A130235_list(m) # G. C. Greubel, Mar 17 2023

Crossrefs

Cf. A000045, A130233, A130234, A130236, A130238, A130240, A130243, A130246, A130244, A130246, A130248, A130251, A130257, A130261.

Sequence in context: A055025 A178805 A088907 * A267531 A219647 A129726

Adjacent sequences: A130232 A130233 A130234 * A130236 A130237 A130238

Keyword

nonn

Author

Hieronymus Fischer, May 17 2007

Status

approved