A112302 - OEIS

%I #104 Feb 16 2025 08:32:59

%S 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,

%T 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,

%U 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

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

%C From _Johannes W. Meijer_, Jun 27 2016: (Start)

%C 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.

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

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

%D S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

%D S. Ramanujan, Collected Papers, Ed. G. H. Hardy et al., AMS Chelsea 2000. See Appendix I. p. 348.

%H Robert G. Wilson v, <a href="/A112302/b112302.txt"> Table of n, a(n) for n = 1..1011 </a>

%H Steven Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, Section 6.10.

%H Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, <a href="https://arxiv.org/abs/2402.09064">On integrality and asymptotic behavior of the (k,l)-Göbel sequences</a>, arXiv:2402.09064 [math.NT], 2024. See p. 2.

%H Olivier Golinelli, <a href="https://arxiv.org/abs/2405.16968">Remote control system of a binary tree of switches - II. balancing for a perfect binary tree</a>, arXiv:2405.16968 [cs.DM], 2024. See p. 17.

%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.jnt.2011.04.010">A note on Somos' quadratic recurrence constant</a>, J. Number Theory 131 (2011), 2061-2063.

%H Dawei Lu and Zexi Song, <a href="http://www.sciencedirect.com/science/article/pii/S0022314X1500133X">Some new continued fraction estimates of the Somos' quadratic recurrence constant</a>, Journal of Number Theory, 155 (2015), 36-45.

%H Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, <a href="https://doi.org/10.1007/s00025-018-0928-0">Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant</a>, Results in Mathematics 74(1) (2019), Article 6.

%H Cristinel Mortici, <a href="http://dx.doi.org/10.1016/j.jnt.2010.06.012">Estimating the Somos' quadratic recurrence constant</a>, J. Number Theory 130 (2010), 2650-1657.

%H Jesús Guillera and Jonathan Sondow, <a href="http://arXiv.org/abs/math.NT/0506319">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, arXiv:math/0506319 [math.NT], 2005-2006; see page 8.

%H Jesús Guillera and Jonathan Sondow, <a href="http://dx.doi.org/10.1007/s11139-007-9102-0">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, Ramanujan J. 16 (2008), 247-270.

%H Jörg Neunhäuserer, <a href="https://arxiv.org/abs/2006.02882">On the universality of Somos' constant</a>, arXiv:2006.02882 [math.DS], 2020.

%H Jonathan Sondow and Petros Hadjicostas, <a href="http://arXiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, arXiv:math/0610499 [math.CA], 2006.

%H Jonathan Sondow and Petros Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl. 332 (2007), 292-314.

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

%H Xu You and Di-Rong Chen, <a href="https://doi.org/10.1016/j.jmaa.2015.12.013">Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant</a>, Mathematical Analysis and Applications, 436(1) (2016), 513-520.

%F Equals Product_{n>=1} n^(1/2^n). - _Jonathan Sondow_, Apr 07 2013

%F Equals exp(A114124) = A188834/2 = sqrt(A259235). - _Hugo Pfoertner_, Nov 23 2024

%e 1.6616879496335941212958189227499507499644186350250682081897111680...

%t RealDigits[ Fold[ N[ Sqrt[ #2*#1], 128] &, Sqrt@ 351, Reverse@ Range@ 350], 10, 111][[1]] (* _Robert G. Wilson v_, Nov 05 2010 *)

%t Exp[-Derivative[1, 0][PolyLog][0, 1/2]] // RealDigits[#, 10, 105]& // First (* _Jean-François Alcover_, Apr 07 2014, after _Jonathan Sondow_ *)

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

%o (PARI) prodinf(n=1,n^2^-n) \\ _Charles R Greathouse IV_, Apr 07 2013

%o (Python)

%o from mpmath import polylog, diff, exp, mp

%o mp.dps = 120

%o somos_const = exp(-diff(lambda n: polylog(n, 1/2), 0))

%o A112302 = [int(d) for d in mp.nstr(somos_const, n=mp.dps)[:-1] if d != '.']  # _Jwalin Bhatt_, Nov 23 2024

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

%Y Cf. A114124 (log).

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

%K cons,nonn,changed

%O 1,2

%A _Michael Somos_, Sep 02 2005