cp's OEIS Frontend

A119812 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>=1} 1/2^A001951(n).

%I A119812 #9 Nov 22 2013 10:13:00
%S A119812 8,5,8,2,6,7,6,5,6,4,6,1,0,0,2,0,5,5,7,9,2,2,6,0,3,0,8,4,3,3,3,7,5,1,
%T A119812 4,8,6,6,4,9,0,5,1,9,0,0,8,3,5,0,6,7,7,8,6,6,7,6,8,4,8,6,7,8,8,7,8,4,
%U A119812 5,5,3,7,9,1,9,1,2,1,1,1,9,5,4,8,7,0,4,9,8,2,7,6,0,6,4,3,1,5,3,1,0,2,5,2
%N A119812 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>=1} 1/2^A001951(n).
%C A119812 Dual constant: A119809 = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n. The binary expansion of this constant is given by A080764 with offset n=1. Plouffe's Inverter describes approximations to this constant as "polylogarithms type of series with the floor function [ ]."
%H A119812 W. W. Adams and J. L. Davison, <a href="http://www.jstor.org/stable/2041889">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.
%H A119812 P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity</a> Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
%e A119812 c = 0.858267656461002055792260308433375148664905190083506778667684867..
%e A119812 Continued fraction (A119813):
%e A119812 c = [0;1,6,18,1032,16777344,288230376151842816,...]
%e A119812 where partial quotients are given by:
%e A119812 PQ[n] = 4^A000129(n-2) + 2^A001333(n-3) (n>2), with PQ[1]=0, PQ[2]=1.
%e A119812 The following are equivalent expressions for the constant:
%e A119812 (1) Sum_{n>=1} A049472(n)/2^n; A049472(n)=[n/sqrt(2)];
%e A119812 (2) Sum_{n>=1} 1/2^A001951(n); A001951(n)=[n*sqrt(2)];
%e A119812 (3) Sum_{n>=1} A080764(n)/2^n; A080764(n)=[(n+1)/sqrt(2)]-[n/sqrt(2)];
%e A119812 where [x] = floor(x).
%e A119812 These series illustrate the above expressions:
%e A119812 (1) c = 0/2^1 + 1/2^2 + 2/2^3 + 2/2^4 + 3/2^5 + 4/2^6 + 4/2^7 +...
%e A119812 (2) c = 1/2^1 + 1/2^2 + 1/2^4 + 1/2^5 + 1/2^7 + 1/2^8 + 1/2^9 +...
%e A119812 (3) c = 1/2^1 + 1/2^2 + 0/2^3 + 1/2^4 + 1/2^5 + 0/2^6 + 1/2^7 +...
%o A119812 (PARI) {a(n)=local(t=sqrt(2)/2,x=sum(m=1,10*n,floor(m*t)/2^m));floor(10^n*x)%10}
%Y A119812 Cf. A119813 (continued fraction), A119814 (convergents); A119809 (dual constant); A000129 (Pell), A001333; Beatty sequences: A049472, A001951, A080764; variants: A014565 (rabbit constant), A073115.
%K A119812 cons,nonn
%O A119812 0,1
%A A119812 _Paul D. Hanna_, May 26 2006
%E A119812 Removed leading zero and corrected offset _R. J. Mathar_, Feb 05 2009