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Records in the continued fraction expansion of Sum_{k >= 2} 2^(-Fibonacci(k)) (A006518).
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%I #8 Sep 04 2023 17:41:18

%S 1,10,14,124,2039,262111,536870655,140737488347135,

%T 75557863725914321321983,10633823966279326983230456465062887423,

%U 803469022129495137770981046170581301261101460862599398686719

%N Records in the continued fraction expansion of Sum_{k >= 2} 2^(-Fibonacci(k)) (A006518).

%C Positions of records are at 2, 7, 9, 13, 19, 27, 35, 47, 63, 87, 117, 159, 217, 301, 413, ...

%F Conjecture: For n>4, a(n) = 2^L(n) - 2^F(n) - 1, with L(n) the Lucas numbers A000032, and F(n) the Fibonacci numbers.

%t Rest[DeleteDuplicates[ContinuedFraction[Sum[2^-Fibonacci[k],{k,2,20}]],GreaterEqual]] (* _Harvey P. Dale_, Sep 04 2023 *)

%o (PARI) m=1;for(n=1,500,t=contfrac(suminf(i=2, 1/2^fibonacci(i)))[n+1];if(m<t,m=t;print1(t,",")))

%K nonn

%O 1,2

%A _Ralf Stephan_, Jan 19 2014