A181313 -id:A181313 - cp's OEIS frontend

A084119 Decimal expansion of the Fibonacci binary number, Sum_{k>0} 1/2^F(k), where F(k) = A000045(k).

1, 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

Offset: 1

Ralf Stephan, May 18 2003

The Fibonacci binary number 1.41027879720... is known to be transcendental.

			1.410278797207865891794043024471063...

Harry J. Smith, Table of n, a(n) for n = 1..20000
David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl 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.
A. J. Van Der Poorten and J. Shallit, A specialised continued fraction, Can. J. Math. 45 (1993), 1067-79.
Index entries for transcendental numbers.

Cf. A000045, A010056, A079586, A181313 (continued fraction), A124091 (essentially the same).

RealDigits[N[Sum[1/2^Fibonacci[k], {k, 1, Infinity}], 120]][[1]] (* Amiram Eldar, Jun 12 2023 *)

suminf(k=1,1/2^fibonacci(k)) \\ This gives the Fibonacci binary number, not the sequence

default(realprecision, 20080); x=suminf(k=1, 1/2^fibonacci(k)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b084119.txt", n, " ", d)); \\

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

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

cp's OEIS Frontend

A084119 Decimal expansion of the Fibonacci binary number, Sum_{k>0} 1/2^F(k), where F(k) = A000045(k).

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

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A125600 Continued fraction expansion of constant defined in A124091.

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