A125600 -id:A125600 - cp's OEIS frontend

A124091 Decimal expansion of Fibonacci binary constant: Sum{i>=0} (1/2)^Fibonacci(i).

2, 4, 1, 0, 2, 7, 8, 7, 9, 7, 2, 0, 7, 8, 6, 5, 8, 9, 1, 7, 9, 4, 0, 4, 3, 0, 2, 4, 4, 7, 1, 0, 6, 3, 1, 4, 4, 4, 8, 3, 4, 2, 3, 9, 2, 4, 5, 9, 5, 2, 7, 8, 7, 7, 2, 5, 9, 3, 2, 9, 2, 4, 6, 7, 9, 3, 0, 0, 7, 3, 5, 1, 6, 8, 2, 6, 0, 2, 7, 9, 4, 5, 3, 5, 1, 6, 1, 2, 3, 3, 0, 1, 2, 1, 4, 5, 9, 0, 2, 3, 3, 2, 8, 5, 1

Offset: 1

R. J. Mathar, Nov 25 2006

This constant is transcendental, see A084119. - Charles R Greathouse IV, Nov 12 2014

			2.4102787972078658917940430244710631444834239245952787725932...

Harry J. Smith, Table of n, a(n) for n = 1..20000
D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, On the binary expansions of algebraic numbers, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
Index entries for transcendental numbers

Cf. A007404 (Kempner-Mahler number), A125600 (continued fraction), A084119 (essentially the same).

Cf. A000301.

RealDigits[ N[ Sum[(1/2)^Fibonacci[i], {i, 0, Infinity}], 111]][[1]] (* Robert G. Wilson v, Nov 26 2006 *)

a=0 ; for(n=0,30, a += .5^fibonacci(n) ; print(a) ; )

default(realprecision, 20080); x=suminf(k=0, 1/2^fibonacci(k)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b124091.txt", n, " ", d)) \\ Harry J. Smith, May 04 2009

Equals Sum_{i>=0} 1/2^A000045(i).

Equals A084119 + 1.

More terms from Robert G. Wilson v, Nov 26 2006

A006518 Continued fraction for Sum_{k >= 2} 2^(-Fibonacci(k)).

Original entry on oeis.org

0, 1, 10, 6, 1, 6, 2, 14, 4, 124, 2, 1, 2, 2039, 1, 9, 1, 1, 1, 262111, 2, 8, 1, 1, 1, 3, 1, 536870655, 4, 16, 3, 1, 3, 7, 1, 140737488347135, 8, 128, 2, 1, 1, 1, 7, 2, 1, 9, 1

Offset: 0

N. J. A. Sloane

			0.91027879720786589179404302... = 0 + 1/(1 + 1/(10 + 1/(6 + 1/(1 + ...)))). - _Harry J. Smith_, May 04 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..564
V. Meally, Letter to N. J. A. Sloane, May 1975
A. J. van der Poorten, Continued fractions of formal power series
A. J. van der Poorten and Jeffrey Shallit, A specialised continued fraction, Canad. J. Math., 45 (1993), 1067-1079.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A000045, A084119, A124091, A125600, A181313.

{ allocatemem(932245000); default(realprecision, 10000); x=suminf(k=2, 1/2^fibonacci(k)); c=contfrac(x); for (n=1, 565, write("b006518.txt", n-1, " ", c[n])); } \\ Harry J. Smith, May 04 2009

Interestingly, a(13)=2^11-2^3-1, a(19)=2^18-2^5-1, a(27)=2^29-2^8-1, a(35)=2^47-2^13-1. - Ralf Stephan, Jun 07 2005

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

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A124091 Decimal expansion of Fibonacci binary constant: Sum{i>=0} (1/2)^Fibonacci(i).

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A006518 Continued fraction for Sum_{k >= 2} 2^(-Fibonacci(k)).

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A181313 Continued fraction expansion of the Fibonacci binary number.

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