A130240 Partial sums of A130239.

0, 2, 4, 6, 9, 12, 15, 18, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260

Offset: 0

Author

Hieronymus Fischer, May 17 2007

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130239, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

Programs

Magma

A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
A130240:= func< n | (&+[A130233(Floor(Sqrt(j))): j in [0..n]]) >;
[A130240(n): n in [0..70]]; // G. C. Greubel, Mar 18 2023

Mathematica

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 + n*Sqrt[5]]];
A130240[n_]:= A130240[n]= Sum[A130233[Floor[Sqrt[j]]], {j,0,n}];
Table[A130240[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

SageMath

def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
def A130240(n): return sum( A130233(floor(sqrt(j))) for j in range(n+1) )
[A130240(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

Formula

a(n) = Sum_{k=0..n} A130239(k).

a(n) = (n+1)*A130233(sqrt(n)) - Fib(A130233(sqrt(n)) + 1) * Fib(A130232(sqrt(n))).

G.f.: (1/(1-x)^2) * Sum_{k>=1} x^(Fib(k)^2).