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

%I A038663 #23 Jul 02 2025 16:01:56
%S A038663 1,3,5,7,9,12,13,16,18,21,22,25,27,29,32,35,36,39,40,43,46,48,49,53,
%T A038663 55,58,60,62,63,67,68,71,73,76,78,81,82,84,87,91,92,96,97,99,102,104,
%U A038663 105,109,110,113,115,118,119,122,125,128,130,132,133,137,138,140,143,146
%N A038663 [ n/F_2 ] + [ n/F_3 ] + [ n/F_4 ] +..., F_n=Fibonacci numbers.
%H A038663 Vaclav Kotesovec, <a href="/A038663/a038663.jpg">Plot of a(n)/n for n = 1..100000</a>
%F A038663 G.f.: (1/(1 - x)) * Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)). - _Ilya Gutkovskiy_, Jul 16 2019
%F A038663 Conjecture: a(n) ~ c * n, where c = A079586 - 1. - _Vaclav Kotesovec_, Aug 30 2021
%e A038663 a(15)=[ 15/1 ]+[ 15/2 ]+[ 15/3 ]+[ 15/5 ]+[ 15/8 ]+[ 15/13 ]+[ 15/21 ]+...=32.
%p A038663 with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k)), k=2..15)) od:
%t A038663 Table[Sum[Floor[n/Fibonacci[k] ],{k,2,200}],{n,70}] (* _Harvey P. Dale_, Jul 21 2021 *)
%t A038663 Table[Sum[Floor[n/Fibonacci[k]], {k, 2, Log[Sqrt[5]*n]/Log[GoldenRatio] + 1}], {n, 1, 100}] (* _Vaclav Kotesovec_, Aug 30 2021 *)
%o A038663 (Magma) [&+[Floor(n/Fibonacci(k+2)):k in [0..n]]:n in [1..64]]; // _Marius A. Burtea_, Jul 16 2019
%Y A038663 Cf. A005086.
%K A038663 nonn,easy
%O A038663 1,2
%A A038663 Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)
%E A038663 More terms from _Simon Plouffe_, who points out that the first differences give A005086
%E A038663 More terms from _James Sellers_, Feb 19 2001