A196530 Decimal expansion of log(2+sqrt(3))/sqrt(3).

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

Offset: 0

Author

R. J. Mathar, Oct 03 2011

Keywords

• nonn
• cons
• easy

Comments

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

Examples

			0.7603459963009463475310942548...
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
• E. D. Krupnikov, K. S. Kolbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95
• R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table in section 2.2, L(m=12,r=4,s=1).
• Index entries for transcendental numbers.

Crossrefs

Cf. A065918, A002194, A110161.

Programs

Mathematica

RealDigits[Log[2 + Sqrt[3]]/Sqrt[3], 10, 89][[1]] (* Bruno Berselli, Dec 20 2011 *)

PARI

log(sqrt(3)+2)/sqrt(3) \\ Charles R Greathouse IV, May 15 2019

Formula

Equals A065918/A002194.

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

Equals Sum_{k>=1} (-1)^(k+1)*2^k/(k * binomial(2*k,k)). - Amiram Eldar, Aug 19 2020

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

Equals A259830 - 2. - Hugo Pfoertner, Apr 06 2024

Equals (1/2)*2F1(1/2,1;3/2;3/4) [Krupnikov] - R. J. Mathar, Jun 11 2024