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A259235 Decimal expansion of sqrt(2*sqrt(3*sqrt(4*...))), a variant of Somos's quadratic recurrence constant.

%I A259235 #26 Feb 16 2025 08:33:25
%S A259235 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 A259235 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 A259235 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 A259235 Decimal expansion of sqrt(2*sqrt(3*sqrt(4*...))), a variant of Somos's quadratic recurrence constant.
%H A259235 G. C. Greubel, <a href="/A259235/b259235.txt">Table of n, a(n) for n = 1..10000</a>
%H A259235 StackExchange, <a href="http://math.stackexchange.com/questions/498774">Improving bound on sqrt(2*sqrt(3*sqrt(4*...)))</a>
%H A259235 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html">Somos's Quadratic Recurrence Constant</a>
%F A259235 Equals A112302^2.
%F A259235 Equals exp( Sum_{n>=1} log(n)/2^(n-1) ).
%F A259235 Also equals exp(-2*PolyLog'(0,1/2)), where PolyLog' is the derivative of PolyLog(n,x) w.r.t. n.
%e A259235 2.7612068419574980332304546465801311048761259807153...
%t A259235 RealDigits[Exp[-2*Derivative[1, 0][PolyLog][0, 1/2]], 10, 102] // First
%t A259235 RealDigits[Exp[2*Sum[(1/2)^n*Log[n], {n, 2, 2000}]], 10, 100][[1]] (* _G. C. Greubel_, Sep 30 2018 *)
%o A259235 (PARI) exp(sumpos(n=1,log(n+1)/2^n)) \\ _Charles R Greathouse IV_, Apr 18 2016
%o A259235 (Magma) SetDefaultRealField(RealField(100)); Exp(2*(&+[(1/2)^n*Log(n): n in [2..2000]])); // _G. C. Greubel_, Sep 30 2018
%Y A259235 Cf. A112302, A114124.
%K A259235 nonn,cons,easy
%O A259235 1,1
%A A259235 _Jean-François Alcover_, Jun 22 2015