A006518 -id:A006518 - cp's OEIS frontend

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

Ralf Stephan, Jan 19 2014

nonn

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

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

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

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.

A181313 Continued fraction expansion of the Fibonacci binary number.

Original entry on oeis.org

1, 2, 2, 3, 2, 28, 1, 13, 1, 2, 1, 123, 1, 6, 1, 2039, 2, 2, 6, 262111, 1, 35, 1, 1, 3, 536870655, 1, 2, 1, 15, 1, 3, 3, 1, 1, 1, 2, 140737488347135, 1, 1, 1, 1, 1, 127, 1, 7, 7, 1, 5, 2, 2, 75557863725914321321983, 1, 1, 2, 5, 1, 2047, 2, 2, 5, 1, 31, 6, 1, 1, 3, 2, 2

Offset: 0

Charles R Greathouse IV, Oct 12 2010

Essentially the same as A125600. - R. J. Mathar, Oct 14 2010

Charles R Greathouse IV, Table of n, a(n) for n = 0..638
D. Bailey, J. Borwein, R. Crandall, and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
J. H. Loxton and A. van der Poorten, Arithmetic properties of certain functions in several variables III, Bulletin of the Australian Mathematical Society, Volume 16, Issue 01, February 1977, pp 15-47.
J. Shallit and A. van der Poorten, A specialised continued fraction, Can. J. Math. 45 (1993), 1067-79.
Alf van der Poorten, In thrall to Fibonacci

Cf. A084119 (decimal expansion), A125600 (essentially the same), A006518.

PARI
```
contfrac(suminf(n=1,2.^-fibonacci(n)))
```

Offset changed by Andrew Howroyd, Aug 09 2024

A125600 Continued fraction expansion of constant defined in A124091.

Original entry on oeis.org

2, 2, 2, 3, 2, 28, 1, 13, 1, 2, 1, 123, 1, 6, 1, 2039, 2, 2, 6, 262111, 1, 35, 1, 1, 3, 536870655, 1, 2, 1, 15, 1, 3, 3, 1, 1, 1, 2, 140737488347135, 1, 1, 1, 1, 1, 127, 1, 7, 7, 1, 5, 2, 2, 75557863725914321321983, 1, 1, 2, 5, 1, 2047, 2, 2, 5, 1, 31, 6, 1, 1, 3, 2, 2

Offset: 0

Robert G. Wilson v, Nov 26 2006

Progressively larger PQ's: 2, 3, 28, 123, 2039, 262111, 536870655, 140737488347135, 75557863725914321321983, 10633823966279326983230456465062887423, 803469022129495137770981046170581301261101460862599398686719, 8543948143683640329580086824678208458410818089426611079788166431288878284152542557401710898184191, ...,.

Cf. A124091 (decimal expansion), A181313 (essentially the same), A006518.

c = N[Sum[(1/2)^Fibonacci[i], {i, 0, Infinity}], 1000]; ContinuedFraction@c

Offset changed by Andrew Howroyd, Aug 03 2024

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

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A181313 Continued fraction expansion of the Fibonacci binary number.

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A125600 Continued fraction expansion of constant defined in A124091.

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