A084119 - OEIS

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

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OFFSET

1,2

COMMENTS

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

LINKS

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.

EXAMPLE

1.410278797207865891794043024471063...

MATHEMATICA

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

PROG

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

(PARI) 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)); \\

CROSSREFS

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

Sequence in context: A096501 A062862 A206799 * A166073 A290724 A283879

Adjacent sequences: A084116 A084117 A084118 * A084120 A084121 A084122

KEYWORD

nonn,cons

AUTHOR

Ralf Stephan, May 18 2003

STATUS

approved