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

%I A130236 #23 Mar 18 2023 03:56:13
%S A130236 0,1,4,8,13,18,24,30,36,43,50,57,64,71,79,87,95,103,111,119,127,135,
%T A130236 144,153,162,171,180,189,198,207,216,225,234,243,252,262,272,282,292,
%U A130236 302,312,322,332,342,352,362,372,382,392,402,412,422,432,442,452,462,473
%N A130236 Partial sums of the 'upper' Fibonacci Inverse A130234.
%H A130236 G. C. Greubel, <a href="/A130236/b130236.txt">Table of n, a(n) for n = 0..5000</a>
%F A130236 a(n) = Sum_{k=0..n} A130234(k).
%F A130236 a(n) = n*A130234(n) - Fibonacci(A130234(n)+1) + 1.
%F A130236 G.f.: (x/(1-x)^2) * Sum_{k>=0} x^Fibonacci(k).
%t A130236 b[n_]:= For[i=0, True, i++, If[Fibonacci[i] >= n, Return[i]]];
%t A130236 b/@ Range[0, 56]//Accumulate (* _Jean-François Alcover_, Apr 13 2020 *)
%o A130236 (Magma)
%o A130236 m:=120;
%o A130236 f:= func< x | x*(&+[x^Fibonacci(j): j in [0..Floor(3*Log(3*m+1))]])/(1-x)^2 >;
%o A130236 R<x>:=PowerSeriesRing(Rationals(), m+1);
%o A130236 [0] cat Coefficients(R!( f(x) )); // _G. C. Greubel_, Mar 18 2023
%o A130236 (SageMath)
%o A130236 m=120
%o A130236 def f(x): return x*sum( x^fibonacci(j) for j in range(1+int(3*log(3*m+1))))/(1-x)^2
%o A130236 def A130236_list(prec):
%o A130236     P.<x> = PowerSeriesRing(ZZ, prec)
%o A130236     return P( f(x) ).list()
%o A130236 A130236_list(m) # _G. C. Greubel_, Mar 18 2023
%Y A130236 Cf. A000045, A130233, A130234, A130235, A130244, A130246, A130244, A130246, A130248, A130252, A130258, A130262.
%K A130236 nonn
%O A130236 0,3
%A A130236 _Hieronymus Fischer_, May 17 2007