login
Decimal expansion of sqrt(2*sqrt(3*sqrt(4*...))), a variant of Somos's quadratic recurrence constant.
2

%I #25 Nov 14 2024 23:22:42

%S 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,

%T 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,

%U 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

%N Decimal expansion of sqrt(2*sqrt(3*sqrt(4*...))), a variant of Somos's quadratic recurrence constant.

%H G. C. Greubel, <a href="/A259235/b259235.txt">Table of n, a(n) for n = 1..10000</a>

%H StackExchange, <a href="http://math.stackexchange.com/questions/498774">Improving bound on sqrt(2*sqrt(3*sqrt(4*...)))</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>

%F Equals A112302^2.

%F Equals exp( Sum_{n>=1} log(n)/2^(n-1) ).

%F Also equals exp(-2*PolyLog'(0,1/2)), where PolyLog' is the derivative of PolyLog(n,x) w.r.t. n.

%e 2.7612068419574980332304546465801311048761259807153...

%t RealDigits[Exp[-2*Derivative[1, 0][PolyLog][0, 1/2]], 10, 102] // First

%t RealDigits[Exp[2*Sum[(1/2)^n*Log[n], {n, 2, 2000}]], 10, 100][[1]] (* _G. C. Greubel_, Sep 30 2018 *)

%o (PARI) exp(sumpos(n=1,log(n+1)/2^n)) \\ _Charles R Greathouse IV_, Apr 18 2016

%o (Magma) SetDefaultRealField(RealField(100)); Exp(2*(&+[(1/2)^n*Log(n): n in [2..2000]])); // _G. C. Greubel_, Sep 30 2018

%Y Cf. A112302, A114124.

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Jun 22 2015