cp's OEIS Frontend

A119809 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.

%I A119809 #8 May 25 2023 08:15:59
%S A119809 2,3,2,2,5,8,8,5,2,2,5,8,8,0,6,7,7,3,0,1,2,1,4,4,0,6,8,2,7,8,7,9,8,4,
%T A119809 0,8,0,1,1,9,5,0,2,5,0,8,0,0,4,3,2,9,2,5,6,6,5,7,1,8,0,6,2,3,9,4,4,0,
%U A119809 5,2,1,7,5,6,0,9,6,9,5,3,9,2,0,6,2,3,5,5,7,5,0,0,7,2,3,9,1,7,7,2,2,4,7,9,7
%N A119809 Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.
%C A119809 Dual constant: A119812 = Sum_{n>=1} A049472(n)/2^n = Sum_{n>=1} 1/2^A001951(n). Since this constant c = 2 + Sum_{n>=1} 1/2^A003151(n), where A003151(n) = n + floor(n*sqrt(2)), then the binary expansion of the fractional part of c has 1's only at positions given by Beatty sequence A003151(n) and zeros elsewhere. Plouffe's Inverter describes approximations to the fractional part of c as "polylogarithms type of series with the floor function [ ]."
%H A119809 Kevin O'Bryant, <a href="https://doi.org/10.1006/jnth.2001.2743">A generating function technique for Beatty sequences and other step sequences</a>, Journal of Number Theory, Volume 94, Issue 2, June 2002, Pages 299-319.
%F A119809 Equals Sum(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < sqrt(2) (O'Bryant, 2002). - _Amiram Eldar_, May 25 2023
%e A119809 c = 2.32258852258806773012144068278798408011950250800432925665718...
%e A119809 Continued fraction (A119810):
%e A119809 c = [2;3,10,132,131104,2199023259648,633825300114114700748888473600,..]
%e A119809 where partial quotients are given by:
%e A119809 PQ(n) = 2^A001333(n-1) + 2^A000129(n-2) (n>1), with PQ(1)=2.
%e A119809 The following are equivalent expressions for the constant:
%e A119809 (1) Sum_{n>=1} 1/2^A049472(n); A049472(n)=[n/sqrt(2)];
%e A119809 (2) Sum_{n>=1} A001951(n)/2^n; A001951(n)=[n*sqrt(2)];
%e A119809 (3) Sum_{n>=1} 1/2^A003151(n) + 2; A003151(n)=[n*sqrt(2)]+n;
%e A119809 (4) Sum_{n>=1} 1/2^A097508(n) - 2; A097508(n)=[n*sqrt(2)]-n;
%e A119809 (5) Sum_{n>=1} A006337(n)/2^n + 1; A006337(n)=[(n+1)*sqrt(2)]-[n*sqrt(2)];
%e A119809 where [x] = floor(x).
%e A119809 These series illustrate the above expressions:
%e A119809 (1) c = 1/2^0 + 1/2^1 + 1/2^2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^4 +...
%e A119809 (2) c = 1/2^1 + 2/2^2 + 4/2^3 + 5/2^4 + 7/2^5 + 8/2^6 + 9/2^7 +...
%e A119809 (3) c = 2 + 1/2^2 + 1/2^4 + 1/2^7 + 1/2^9 + 1/2^12 + 1/2^14 +...
%e A119809 (4) c =-2 + 1/2^0 + 1/2^0 + 1/2^1 + 1/2^1 + 1/2^2 + 1/2^2 + 1/2^2 +...
%e A119809 (5) c = 1 + 1/2^1 + 2/2^2 + 1/2^3 + 2/2^4 + 1/2^5 + 1/2^6 + 2/2^7 +...
%o A119809 (PARI) {a(n)=local(t=sqrt(2),x=sum(m=1,10*n,floor(m*t)/2^m));floor(10^n*x)%10}
%Y A119809 Cf. A119810 (continued fraction), A119811 (convergents); A119812 (dual constant); A000129 (Pell), A001333; Beatty sequences: A049472, A001951, A003151, A097508, A006337; variants: A014565 (rabbit constant), A073115.
%Y A119809 Cf. A081544, A081550, A081573.
%K A119809 cons,nonn
%O A119809 1,1
%A A119809 _Paul D. Hanna_, May 26 2006