A065446 Decimal expansion of Product_{k>=1} (1-1/2^k)^(-1).
3, 4, 6, 2, 7, 4, 6, 6, 1, 9, 4, 5, 5, 0, 6, 3, 6, 1, 1, 5, 3, 7, 9, 5, 7, 3, 4, 2, 9, 2, 4, 4, 3, 1, 1, 6, 4, 5, 4, 0, 7, 5, 7, 9, 0, 2, 9, 0, 4, 4, 3, 8, 3, 9, 1, 3, 2, 9, 3, 5, 3, 0, 3, 1, 7, 5, 8, 9, 1, 5, 4, 3, 9, 7, 4, 0, 4, 2, 0, 6, 4, 5, 6, 8, 7, 9, 2, 7, 7, 4, 0, 2, 9, 4, 8, 4, 3, 3, 5, 3, 5, 0, 8, 8, 0
Offset: 1
Examples
3.46274661945506361153795734292443116454075790290...
References
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
Links
- Harry J. Smith, Table of n, a(n) for n=1..2000
- Steven R. Finch, Digital Search Tree Constants [Broken link]
- Steven R. Finch, Digital Search Tree Constants [From the Wayback machine]
- Igor Pak, Greta Panova, and Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018. [p. 13]
- Jonathan Sondow and Eric W. Weisstein, MathWorld: Wallis Formula.
Programs
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Maple
evalf(1+sum(2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 22 2020
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Mathematica
N[ Product[ 1/(1 - 1/2^k), {k, 1, Infinity} ], 500 ] RealDigits[1/QPochhammer[1/2, 1/2], 10, 100][[1]] (* Vaclav Kotesovec, Jun 22 2014 *) -
PARI
{ default(realprecision, 2080); x=prodinf(k=1, 1/(1 - 1/2^k)); for (n=1, 2000, d=floor(x); x=(x-d)*10; write("b065446.txt", n, " ", d)) } \\ Harry J. Smith, Oct 19 2009 -
PARI
prodinf(k=1, 1/(1-1/2^k)) \\ Michel Marcus, Feb 22 2020
Formula
Equals Sum_{n>=0} 1/A002884(n)*Product_{j=1..n} 2^j/(2^j-1). - Geoffrey Critzer, Jun 30 2017
Equals 1/QPochhammer(1/2, 1/2){infinity}. - _G. C. Greubel, Jan 18 2018
Equals 1 + Sum_{n>=1} 2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). - Robert FERREOL, Feb 22 2020
Equals 1 / A048651 (constant). - Hugo Pfoertner, Nov 28 2020
Equals Sum_{n>=0} A000041(n)/2^n. - Amiram Eldar, Jan 19 2021
Extensions
More terms from Robert G. Wilson v, Nov 19 2001
Further terms from Vladeta Jovovic, Dec 01 2001