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A084909 Indices of Fibonacci numbers that satisfy: Sum_{k>=1} 1/F(a(k)) = tau-1, where F(k) gives the k-th Fibonacci number with F(0)=1, such that the partial sums are nearest to, but never exceed, tau-1 = (sqrt(5)-1)/2. 3

%I

%S 2,6,8,10,17,21,24,26,29,34,42,46,49,51,54,56,59,61,63,65,67,69,71,77,

%T 79,81,85,88,90,92,94,101,103,110,114,121,123,129,131,134,137,143,146,

%U 149,154,156,160,163,167,172,176,181,183,185,190,193,196,199,204,206

%N Indices of Fibonacci numbers that satisfy: Sum_{k>=1} 1/F(a(k)) = tau-1, where F(k) gives the k-th Fibonacci number with F(0)=1, such that the partial sums are nearest to, but never exceed, tau-1 = (sqrt(5)-1)/2.

%C Corresponding Fibonacci numbers are given by A084910.

%H Amiram Eldar, <a href="/A084909/b084909.txt">Table of n, a(n) for n = 1..10000</a>

%e (sqrt(5)-1)/2 = 1/F(2) + 1/F(6) + 1/F(8) + 1/F(10) + 1/F(17) + 1/F(21) + ... = 1/2 + 1/13 + 1/34 + 1/89 + 1/2584 + 1/17711 + 1/75025 +...

%t seq = {}; s = GoldenRatio - 1; m = 3; Do[AppendTo[seq, m - 1]; s -= (1/Fibonacci[m]); While[Fibonacci[m] <= 1/s, m++], {60}]; seq (* _Amiram Eldar_, Nov 01 2019 *)

%o (PARI) x=(sqrt(5)-1)/2; a=2; S=0; for(n=1,100,b=a+1; while(abs(S+1/fibonacci(b))>x,b++); S=S+1/fibonacci(b); a=b; print1(b-1,","))

%Y Cf. A000045, A001622, A084907, A084910.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Jun 10 2003

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Last modified March 7 14:00 EST 2021. Contains 341886 sequences. (Running on oeis4.)