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

1,2

Comments

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

Links

Table of n, a(n) for n=1..11.

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.

Mathematica

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

Prog

(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, ", ")))

Crossrefs

Sequence in context: A254003 A219682 A167334 * A246769 A322566 A004475

Adjacent sequences: A236142 A236143 A236144 * A236146 A236147 A236148

Keyword

nonn

Author

Ralf Stephan, Jan 19 2014

Status

approved