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%I #12 Nov 01 2019 18:34:57
%S 2,13,34,89,2584,17711,75025,196418,832040,9227465,433494437,
%T 2971215073,12586269025,32951280099,139583862445,365435296162,
%U 1548008755920,4052739537881,10610209857723,27777890035288,72723460248141
%N Fibonacci numbers that satisfy: Sum_{k>=1} 1/a(k) = tau-1, such that the partial sums are nearest to, but never exceed, tau-1 = (sqrt(5)-1)/2.
%C Corresponding Fibonacci indices are given by A084909.
%H Amiram Eldar, <a href="/A084910/b084910.txt">Table of n, a(n) for n = 1..1323</a>
%F a(n) = A000045(A084909(n)+1). - _Amiram Eldar_, Nov 01 2019
%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, Fibonacci[m]]; s -= (1/Fibonacci[m]); While[Fibonacci[m] <= 1/s, m++], {21}]; 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(fibonacci(b),","))
%Y Cf. A000045, A001622, A084908, A084909.
%K nonn
%O 1,1
%A _Paul D. Hanna_, Jun 10 2003
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