login
A130240
Partial sums of A130239.
14
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
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
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, May 17 2007
STATUS
approved