A112302 Decimal expansion of quadratic recurrence constant sqrt(1 * sqrt(2 * sqrt(3 * sqrt(4 * ...)))).

1, 6, 6, 1, 6, 8, 7, 9, 4, 9, 6, 3, 3, 5, 9, 4, 1, 2, 1, 2, 9, 5, 8, 1, 8, 9, 2, 2, 7, 4, 9, 9, 5, 0, 7, 4, 9, 9, 6, 4, 4, 1, 8, 6, 3, 5, 0, 2, 5, 0, 6, 8, 2, 0, 8, 1, 8, 9, 7, 1, 1, 1, 6, 8, 0, 2, 5, 6, 0, 9, 0, 2, 9, 8, 2, 6, 3, 8, 3, 7, 2, 7, 9, 0, 8, 3, 6, 9, 1, 7, 6, 4, 1, 1, 4, 6, 1, 1, 6, 7, 1, 5, 5, 2, 8

Offset: 1

Author

Michael Somos, Sep 02 2005

Keywords

• cons
• nonn

Comments

From Johannes W. Meijer, Jun 27 2016: (Start)

With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.

For similar formulas, see A090998 and A135002. For background information, see A274181.

The structure of the n! * LP(n) formulas leads to the multinomial coefficients A036039. (End)

Examples

			1.6616879496335941212958189227499507499644186350250682081897111680...
		

References

• S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.
• S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., AMS Chelsea 2000. See Appendix I. p. 348.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1011
• Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, Section 6.10.
• Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 2.
• Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
• M. D. Hirschhorn, A note on Somos' quadratic recurrence constant, J. Number Theory 131 (2011), 2061-2063.
• Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, 155 (2015), 36-45.
• Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.
• Cristinel Mortici, Estimating the Somos' quadratic recurrence constant, J. Number Theory 130 (2010), 2650-1657.
• Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006; see page 8.
• Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008), 247-270.
• Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.
• Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
• Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
• Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
• Wikipedia, Somos' quadratic recurrence constant
• Xu You and Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

Crossrefs

Cf. A052129, A055209, A116603, A123851, A123852, A123853, A123854.

Cf. A114124 (log).

Cf. A036039, A090998, A135002, A188834, A259235, A274181.

Programs

Mathematica

RealDigits[ Fold[ N[ Sqrt[ #2*#1], 128] &, Sqrt@ 351, Reverse@ Range@ 350], 10, 111][[1]] (* Robert G. Wilson v, Nov 05 2010 *)
Exp[-Derivative[1, 0][PolyLog][0, 1/2]] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Apr 07 2014, after Jonathan Sondow *)

PARI

{a(n) = if( n<-1, 0, n++; default( realprecision, n+2); floor( prodinf( k=1, k^2^-k)* 10^n) % 10)};

PARI

prodinf(n=1,n^2^-n) \\ Charles R Greathouse IV, Apr 07 2013

Python

from mpmath import polylog, diff, exp, mp
mp.dps = 120
somos_const = exp(-diff(lambda n: polylog(n, 1/2), 0))
A112302 = [int(d) for d in mp.nstr(somos_const, n=mp.dps)[:-1] if d != '.']  # Jwalin Bhatt, Nov 23 2024

Formula

Equals Product_{n>=1} n^(1/2^n). - Jonathan Sondow, Apr 07 2013

Equals exp(A114124) = A188834/2 = sqrt(A259235). - Hugo Pfoertner, Nov 23 2024

From Jwalin Bhatt, Apr 02 2025: (Start)

Equals exp(-PolyLog'(0,1/2)), where PolyLog'(x,y) represents the derivative of the polylogarithm w.r.t. x.

Equals Product_{n>=1} (1+1/n)^(1/2^n).

Equals exp(Sum_{n>=2} log(n)/2^n).

Equals 2*exp(Sum_{n>=1} (log(1+1/n)-1/n)/2^n). (End)