A196530 - OEIS

%I #43 Jun 11 2024 10:56:31

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

%T 2,4,2,0,1,6,2,7,7,3,0,9,4,7,1,7,6,4,2,7,0,2,0,5,7,0,6,7,0,2,6,0,0,5,

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

%N Decimal expansion of log(2+sqrt(3))/sqrt(3).

%C Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=1.

%D L. B. W. Jolley, Summation of series, Dover (1961), eq. (83), page 16.

%H Vincenzo Librandi, <a href="/A196530/b196530.txt">Table of n, a(n) for n = 0..10000</a>

%H Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.

%H E. D. Krupnikov, K. S. Kolbig, <a href="https://dx.doi.org/10.1016/S0377-0427(96)00111-2">Some special cases of the generalized hypergeometric function (q+1)Fq</a>, J. Comp. Appl. Math. 78 (1997) 79-95

%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, Table in section 2.2, L(m=12,r=4,s=1).

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals A065918/A002194.

%F Equals Sum_{n>=1} A110161(n)/n.

%F Equals Sum_{k>=1} (-1)^(k+1)*2^k/(k * binomial(2*k,k)). - _Amiram Eldar_, Aug 19 2020

%F Equals 1/Product_{p prime} (1 - Kronecker(12,p)/p), where Kronecker(12,p) = 0 if p = 2 or 3, 1 if p == 1 or 11 (mod 12) or -1 if p == 5 or 7 (mod 12). - _Amiram Eldar_, Dec 17 2023

%F Equals A259830 - 2. - _Hugo Pfoertner_, Apr 06 2024

%F Equals (1/2)*2F1(1/2,1;3/2;3/4) [Krupnikov] - _R. J. Mathar_, Jun 11 2024

%e 0.7603459963009463475310942548...

%t RealDigits[Log[2 + Sqrt[3]]/Sqrt[3], 10, 89][[1]] (* _Bruno Berselli_, Dec 20 2011 *)

%o (PARI) log(sqrt(3)+2)/sqrt(3) \\ _Charles R Greathouse IV_, May 15 2019

%Y Cf. A065918, A002194, A110161.

%K nonn,cons,easy

%O 0,1

%A _R. J. Mathar_, Oct 03 2011