A038663 [ n/F_2 ] + [ n/F_3 ] + [ n/F_4 ] +..., F_n=Fibonacci numbers.

1, 3, 5, 7, 9, 12, 13, 16, 18, 21, 22, 25, 27, 29, 32, 35, 36, 39, 40, 43, 46, 48, 49, 53, 55, 58, 60, 62, 63, 67, 68, 71, 73, 76, 78, 81, 82, 84, 87, 91, 92, 96, 97, 99, 102, 104, 105, 109, 110, 113, 115, 118, 119, 122, 125, 128, 130, 132, 133, 137, 138, 140, 143, 146

Offset: 1

Author

Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

Keywords

• nonn
• easy

Examples

			a(15)=[ 15/1 ]+[ 15/2 ]+[ 15/3 ]+[ 15/5 ]+[ 15/8 ]+[ 15/13 ]+[ 15/21 ]+...=32.
		

Links

• Vaclav Kotesovec, Plot of a(n)/n for n = 1..100000

Crossrefs

Cf. A005086.

Programs

Magma

[&+[Floor(n/Fibonacci(k+2)):k in [0..n]]:n in [1..64]]; // Marius A. Burtea, Jul 16 2019

Maple

with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k)), k=2..15)) od:

Mathematica

Table[Sum[Floor[n/Fibonacci[k] ],{k,2,200}],{n,70}] (* Harvey P. Dale, Jul 21 2021 *)
Table[Sum[Floor[n/Fibonacci[k]], {k, 2, Log[Sqrt[5]*n]/Log[GoldenRatio] + 1}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 30 2021 *)

Formula

G.f.: (1/(1 - x)) * Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Jul 16 2019

Conjecture: a(n) ~ c * n, where c = A079586 - 1. - Vaclav Kotesovec, Aug 30 2021

Extensions

More terms from Simon Plouffe, who points out that the first differences give A005086

More terms from James Sellers, Feb 19 2001