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A130240 Partial sums of A130239.

%I A130240 #14 Mar 18 2023 18:22:51
%S A130240 0,2,4,6,9,12,15,18,21,25,29,33,37,41,45,49,53,57,61,65,69,73,77,81,
%T A130240 85,90,95,100,105,110,115,120,125,130,135,140,145,150,155,160,165,170,
%U A130240 175,180,185,190,195,200,205,210,215,220,225,230,235,240,245,250,255,260
%N A130240 Partial sums of A130239.
%H A130240 G. C. Greubel, <a href="/A130240/b130240.txt">Table of n, a(n) for n = 0..5000</a>
%F A130240 a(n) = Sum_{k=0..n} A130239(k).
%F A130240 a(n) = (n+1)*A130233(sqrt(n)) - Fib(A130233(sqrt(n)) + 1) * Fib(A130232(sqrt(n))).
%F A130240 G.f.: (1/(1-x)^2) * Sum_{k>=1} x^(Fib(k)^2).
%t A130240 A130233[n_]:= Floor[Log[GoldenRatio, 3/2 + n*Sqrt[5]]];
%t A130240 A130240[n_]:= A130240[n]= Sum[A130233[Floor[Sqrt[j]]], {j,0,n}];
%t A130240 Table[A130240[n], {n,0,70}] (* _G. C. Greubel_, Mar 18 2023 *)
%o A130240 (Magma)
%o A130240 A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
%o A130240 A130240:= func< n | (&+[A130233(Floor(Sqrt(j))): j in [0..n]]) >;
%o A130240 [A130240(n): n in [0..70]]; // _G. C. Greubel_, Mar 18 2023
%o A130240 (SageMath)
%o A130240 def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
%o A130240 def A130240(n): return sum( A130233(floor(sqrt(j))) for j in range(n+1) )
%o A130240 [A130240(n) for n in range(71)] # _G. C. Greubel_, Mar 18 2023
%Y A130240 Cf. A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130239, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.
%K A130240 nonn
%O A130240 0,2
%A A130240 _Hieronymus Fischer_, May 17 2007