A236145 Records in the continued fraction expansion of Sum_{k >= 2} 2^(-Fibonacci(k)) (A006518).

1, 10, 14, 124, 2039, 262111, 536870655, 140737488347135, 75557863725914321321983, 10633823966279326983230456465062887423, 803469022129495137770981046170581301261101460862599398686719

Offset: 1

Author

Ralf Stephan, Jan 19 2014

Keywords

• nonn

Comments

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

Programs

Mathematica

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

PARI

m=1;for(n=1,500,t=contfrac(suminf(i=2, 1/2^fibonacci(i)))[n+1];if(m
				

Formula

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.