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.

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, 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, 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

Offset: 1

Author

Paul D. Hanna, May 26 2006

Keywords

• cons
• nonn

Comments

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 [ ]."

Examples

			c = 2.32258852258806773012144068278798408011950250800432925665718...
Continued fraction (A119810):
c = [2;3,10,132,131104,2199023259648,633825300114114700748888473600,..]
where partial quotients are given by:
PQ(n) = 2^A001333(n-1) + 2^A000129(n-2) (n>1), with PQ(1)=2.
The following are equivalent expressions for the constant:
(1) Sum_{n>=1} 1/2^A049472(n); A049472(n)=[n/sqrt(2)];
(2) Sum_{n>=1} A001951(n)/2^n; A001951(n)=[n*sqrt(2)];
(3) Sum_{n>=1} 1/2^A003151(n) + 2; A003151(n)=[n*sqrt(2)]+n;
(4) Sum_{n>=1} 1/2^A097508(n) - 2; A097508(n)=[n*sqrt(2)]-n;
(5) Sum_{n>=1} A006337(n)/2^n + 1; A006337(n)=[(n+1)*sqrt(2)]-[n*sqrt(2)];
where [x] = floor(x).
These series illustrate the above expressions:
(1) c = 1/2^0 + 1/2^1 + 1/2^2 + 1/2^2 + 1/2^3 + 1/2^4 + 1/2^4 +...
(2) c = 1/2^1 + 2/2^2 + 4/2^3 + 5/2^4 + 7/2^5 + 8/2^6 + 9/2^7 +...
(3) c = 2 + 1/2^2 + 1/2^4 + 1/2^7 + 1/2^9 + 1/2^12 + 1/2^14 +...
(4) c =-2 + 1/2^0 + 1/2^0 + 1/2^1 + 1/2^1 + 1/2^2 + 1/2^2 + 1/2^2 +...
(5) c = 1 + 1/2^1 + 2/2^2 + 1/2^3 + 2/2^4 + 1/2^5 + 1/2^6 + 2/2^7 +...
		

Links

• Kevin O'Bryant, A generating function technique for Beatty sequences and other step sequences, Journal of Number Theory, Volume 94, Issue 2, June 2002, Pages 299-319.

Crossrefs

Cf. A119810 (continued fraction), A119811 (convergents); A119812 (dual constant); A000129 (Pell), A001333; Beatty sequences: A049472, A001951, A003151, A097508, A006337; variants: A014565 (rabbit constant), A073115.

Cf. A081544, A081550, A081573.

Programs

PARI

{a(n)=local(t=sqrt(2),x=sum(m=1,10*n,floor(m*t)/2^m));floor(10^n*x)%10}

Formula

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