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

0,2

Links

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

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).

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 *)

Prog

(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
(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
(Python)
from math import isqrt
def A130240(n):
    a, b, c, m = 0, 1, 0, isqrt(n)
    while b <= m:
        c += n+1
        a, b = b, a+b
    return c-a*b # Chai Wah Wu, Sep 07 2025

Crossrefs

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

Sequence in context: A014011 A064424 A067850 * A143118 A162800 A083652

Adjacent sequences: A130237 A130238 A130239 * A130241 A130242 A130243

Keyword

nonn

Author

Hieronymus Fischer, May 17 2007

Status

approved