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A119812
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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).
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6
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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, 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, 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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| 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 [ ]."
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EXAMPLE
| c = 0.858267656461002055792260308433375148664905190083506778667684867..
Continued fraction (A119813):
c = [0;1,6,18,1032,16777344,288230376151842816,...]
where partial quotients are given by:
PQ[n] = 4^A000129(n-2) + 2^A001333(n-3) (n>2), with PQ[1]=0, PQ[2]=1.
The following are equivalent expressions for the constant:
(1) Sum_{n>=1} A049472(n)/2^n; A049472(n)=[n/sqrt(2)];
(2) Sum_{n>=1} 1/2^A001951(n); A001951(n)=[n*sqrt(2)];
(3) Sum_{n>=1} A080764(n)/2^n; A080764(n)=[(n+1)/sqrt(2)]-[n/sqrt(2)];
where [x] = floor(x).
These series illustrate the above expressions:
(1) c = 0/2^1 + 1/2^2 + 2/2^3 + 2/2^4 + 3/2^5 + 4/2^6 + 4/2^7 +...
(2) c = 1/2^1 + 1/2^2 + 1/2^4 + 1/2^5 + 1/2^7 + 1/2^8 + 1/2^9 +...
(3) c = 1/2^1 + 1/2^2 + 0/2^3 + 1/2^4 + 1/2^5 + 0/2^6 + 1/2^7 +...
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PROG
| (PARI) {a(n)=local(t=sqrt(2)/2, x=sum(m=1, 10*n, floor(m*t)/2^m)); floor(10^n*x)%10}
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CROSSREFS
| Cf. A119813 (continued fraction), A119814 (convergents); A119809 (dual constant); A000129 (Pell), A001333; Beatty sequences: A049472, A001951, A080764; variants: A014565 (rabbit constant), A073115.
Sequence in context: A117039 A085663 A021925 * A153799 A086235 A157742
Adjacent sequences: A119809 A119810 A119811 * A119813 A119814 A119815
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KEYWORD
| cons,nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), May 26 2006
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EXTENSIONS
| Removed leading zero and corrected offset R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2009
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