A130235 - OEIS

%I #19 Mar 17 2023 17:07:52

%S 0,2,5,9,13,18,23,28,34,40,46,52,58,65,72,79,86,93,100,107,114,122,

%T 130,138,146,154,162,170,178,186,194,202,210,218,227,236,245,254,263,

%U 272,281,290,299,308,317,326,335,344,353,362,371,380,389,398,407,417,427

%N Partial sums of the 'lower' Fibonacci Inverse A130233.

%H G. C. Greubel, <a href="/A130235/b130235.txt">Table of n, a(n) for n = 0..5000</a>

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

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

%t 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 *)

%o (Magma)

%o m:=120;

%o f:= func< x | (&+[x^Fibonacci(j): j in [1..Floor(3*Log(3*m+1))]])/(1-x)^2 >;

%o R<x>:=PowerSeriesRing(Rationals(), m+1);

%o [0] cat Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 17 2023

%o (SageMath)

%o m=120

%o def f(x): return sum( x^fibonacci(j) for j in range(1, int(3*log(3*m+1))))/(1-x)^2

%o def A130235_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( f(x) ).list()

%o A130235_list(m) # _G. C. Greubel_, Mar 17 2023

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

%K nonn

%O 0,2

%A _Hieronymus Fischer_, May 17 2007