A259235 Decimal expansion of sqrt(2*sqrt(3*sqrt(4*...))), a variant of Somos's quadratic recurrence constant.
2, 7, 6, 1, 2, 0, 6, 8, 4, 1, 9, 5, 7, 4, 9, 8, 0, 3, 3, 2, 3, 0, 4, 5, 4, 6, 4, 6, 5, 8, 0, 1, 3, 1, 1, 0, 4, 8, 7, 6, 1, 2, 5, 9, 8, 0, 7, 1, 5, 3, 0, 4, 8, 5, 0, 9, 5, 0, 7, 4, 5, 9, 6, 1, 3, 7, 5, 5, 9, 5, 5, 9, 1, 9, 4, 3, 9, 2, 7, 1, 5, 8, 3, 4, 8, 0, 1, 7, 2, 6, 6, 3, 0, 8, 9, 8, 9, 4, 4, 3, 4, 1
Offset: 1
Examples
2.7612068419574980332304546465801311048761259807153...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..10000
- StackExchange, Improving bound on sqrt(2*sqrt(3*sqrt(4*...)))
- Eric Weisstein's MathWorld, Somos's Quadratic Recurrence Constant
Programs
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Magma
SetDefaultRealField(RealField(100)); Exp(2*(&+[(1/2)^n*Log(n): n in [2..2000]])); // G. C. Greubel, Sep 30 2018
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Mathematica
RealDigits[Exp[-2*Derivative[1, 0][PolyLog][0, 1/2]], 10, 102] // First RealDigits[Exp[2*Sum[(1/2)^n*Log[n], {n, 2, 2000}]], 10, 100][[1]] (* G. C. Greubel, Sep 30 2018 *) -
PARI
exp(sumpos(n=1,log(n+1)/2^n)) \\ Charles R Greathouse IV, Apr 18 2016
Formula
Equals A112302^2.
Equals exp( Sum_{n>=1} log(n)/2^(n-1) ).
Also equals exp(-2*PolyLog'(0,1/2)), where PolyLog' is the derivative of PolyLog(n,x) w.r.t. n.