cp's OEIS Frontend

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

%I A084119 #38 Jun 12 2023 02:55:10
%S A084119 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,
%T A084119 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,
%U A084119 7,3,5,1,6,8,2,6,0,2,7,9,4,5,3,5,1,6,1,2,3,3
%N A084119 Decimal expansion of the Fibonacci binary number, Sum_{k>0} 1/2^F(k), where F(k) = A000045(k).
%C A084119 The Fibonacci binary number 1.41027879720... is known to be transcendental.
%H A084119 Harry J. Smith, <a href="/A084119/b084119.txt">Table of n, a(n) for n = 1..20000</a>
%H A084119 David H. Bailey, Jonathan M. Borwein, Richard E. Crandall, and Carl Pomerance, <a href="https://doi.org/10.5802/jtnb.457">On the binary expansions of algebraic numbers</a>, Journal de Théorie des Nombres de Bordeaux 16 (2004), 487-518.
%H A084119 J. H. Loxton and A. van der Poorten, <a href="http://dx.doi.org/10.1017/S0004972700022978">Arithmetic properties of certain functions in several variables III</a>, Bulletin of the Australian Mathematical Society, Volume 16, Issue 01, February 1977, pp 15-47.
%H A084119 A. J. Van Der Poorten and J. Shallit, <a href="http://dx.doi.org/10.4153/CJM-1993-058-5">A specialised continued fraction</a>, Can. J. Math. 45 (1993), 1067-79.
%H A084119 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A084119 1.410278797207865891794043024471063...
%t A084119 RealDigits[N[Sum[1/2^Fibonacci[k], {k, 1, Infinity}], 120]][[1]] (* _Amiram Eldar_, Jun 12 2023 *)
%o A084119 (PARI) suminf(k=1,1/2^fibonacci(k)) \\ This gives the Fibonacci binary number, not the sequence
%o A084119 (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)); \\
%Y A084119 Cf. A000045, A010056, A079586, A181313 (continued fraction), A124091 (essentially the same).
%K A084119 nonn,cons
%O A084119 1,2
%A A084119 _Ralf Stephan_, May 18 2003