A130236 Partial sums of the 'upper' Fibonacci Inverse A130234.

0, 1, 4, 8, 13, 18, 24, 30, 36, 43, 50, 57, 64, 71, 79, 87, 95, 103, 111, 119, 127, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 262, 272, 282, 292, 302, 312, 322, 332, 342, 352, 362, 372, 382, 392, 402, 412, 422, 432, 442, 452, 462, 473

Offset: 0

Author

Hieronymus Fischer, May 17 2007

Keywords

• nonn

Links

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

Crossrefs

Cf. A000045, A130233, A130234, A130235, A130244, A130246, A130244, A130246, A130248, A130252, A130258, A130262.

Programs

Magma

m:=120;
f:= func< x | x*(&+[x^Fibonacci(j): j in [0..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
R:=PowerSeriesRing(Rationals(), m+1);
[0] cat Coefficients(R!( f(x) )); // G. C. Greubel, Mar 18 2023

Mathematica

b[n_]:= For[i=0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
b/@ Range[0, 56]//Accumulate (* Jean-François Alcover, Apr 13 2020 *)

SageMath

m=120
def f(x): return x*sum( x^fibonacci(j) for j in range(1+int(3*log(3*m+1))))/(1-x)^2
def A130236_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A130236_list(m) # G. C. Greubel, Mar 18 2023

Formula

a(n) = Sum_{k=0..n} A130234(k).

a(n) = n*A130234(n) - Fibonacci(A130234(n)+1) + 1.

G.f.: (x/(1-x)^2) * Sum_{k>=0} x^Fibonacci(k).