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

%I A130235 #19 Mar 17 2023 17:07:52
%S A130235 0,2,5,9,13,18,23,28,34,40,46,52,58,65,72,79,86,93,100,107,114,122,
%T A130235 130,138,146,154,162,170,178,186,194,202,210,218,227,236,245,254,263,
%U A130235 272,281,290,299,308,317,326,335,344,353,362,371,380,389,398,407,417,427
%N A130235 Partial sums of the 'lower' Fibonacci Inverse A130233.
%H A130235 G. C. Greubel, <a href="/A130235/b130235.txt">Table of n, a(n) for n = 0..5000</a>
%F A130235 a(n) = Sum_{k=0..n} A130233(k) = (n+1)*A130233(n) - Fib(A130233(n)+2) + 1.
%F A130235 G.f.: 1/(1-x)^2 * Sum_{k>=1} x^Fib(k). [corrected by _Joerg Arndt_, Apr 14 2020]
%t A130235 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 A130235 (Magma)
%o A130235 m:=120;
%o A130235 f:= func< x | (&+[x^Fibonacci(j): j in [1..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
%o A130235 R<x>:=PowerSeriesRing(Rationals(), m+1);
%o A130235 [0] cat Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 17 2023
%o A130235 (SageMath)
%o A130235 m=120
%o A130235 def f(x): return sum( x^fibonacci(j) for j in range(1, int(3*log(3*m+1))))/(1-x)^2
%o A130235 def A130235_list(prec):
%o A130235     P.<x> = PowerSeriesRing(ZZ, prec)
%o A130235     return P( f(x) ).list()
%o A130235 A130235_list(m) # _G. C. Greubel_, Mar 17 2023
%Y A130235 Cf. A000045, A130233, A130234, A130236, A130238, A130240, A130243, A130246, A130244, A130246, A130248, A130251, A130257, A130261.
%K A130235 nonn
%O A130235 0,2
%A A130235 _Hieronymus Fischer_, May 17 2007