A130236 - OEIS

%I #23 Mar 18 2023 03:56:13

%S 0,1,4,8,13,18,24,30,36,43,50,57,64,71,79,87,95,103,111,119,127,135,

%T 144,153,162,171,180,189,198,207,216,225,234,243,252,262,272,282,292,

%U 302,312,322,332,342,352,362,372,382,392,402,412,422,432,442,452,462,473

%N Partial sums of the 'upper' Fibonacci Inverse A130234.

%H G. C. Greubel, <a href="/A130236/b130236.txt">Table of n, a(n) for n = 0..5000</a>

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

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

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

%t b[n_]:= For[i=0, True, i++, If[Fibonacci[i] >= n, Return[i]]];

%t b/@ Range[0, 56]//Accumulate (* _Jean-François Alcover_, Apr 13 2020 *)

%o (Magma)

%o m:=120;

%o f:= func< x | x*(&+[x^Fibonacci(j): j in [0..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 18 2023

%o (SageMath)

%o m=120

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

%o def A130236_list(prec):

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

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

%o A130236_list(m) # _G. C. Greubel_, Mar 18 2023

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

%K nonn

%O 0,3

%A _Hieronymus Fischer_, May 17 2007