A086466 Decimal expansion of 2*sqrt(5)/5 arccsch(2).

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

Offset: 0

Author

Eric W. Weisstein, Jul 21 2003

Keywords

• nonn
• cons

Comments

Equals the value of the Dirichlet L-series of the non-principal character modulo 5 (A080891) at s=1. - Jianing Song, Nov 16 2019

Examples

			0.43040894096400403888943323295060542542457...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7.

Links

• R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22 for L(m=5,r=3,s=1).
• H.-J. Seiffert, Problem B-771, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; More Sums, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
• Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
• Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
• Index entries for transcendental numbers

Crossrefs

Cf. A086465, A086467, A086468.

Cf. A000032, A000045, A080891, A328717 (s=2), A344041.

Programs

Mathematica

2*Log[GoldenRatio]/Sqrt[5] // RealDigits[#, 10, 102]& // First (* Jean-François Alcover, Apr 18 2014 *)

PARI

2*log((1+sqrt(5))/2)/sqrt(5) \\ Stefano Spezia, Oct 15 2024

Formula

Equals Sum_{k>=1} (-1)^(k-1)/(k*binomial(2*k,k)).

Equals A010532 * A002390 / 10. - R. J. Mathar, Jul 26 2010

Also equals f'(0) = 2*log(phi)/sqrt(5), with f(x) = (phi^x-cos(Pi*x)*phi^-x)/sqrt(5), the real Fibonacci interpolating function. - Jean-François Alcover, Apr 04 2014

Equals Sum_{k>=1} A080891(k)/k = Sum_{k>=1} Kronecker(5,k)/k = 1 - 1/2 - 1/3 + 1/4 + 1/6 - 1/7 - 1/8 + 1/9 + ... - Jianing Song, Nov 16 2019

Equals Sum_{k>=1} F(k)/(k*2^(k+1)), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Aug 10 2020

Sum_{k>=1} (2*k+1)*Lucas(k)/(k*(k+1)*2^k) = 10*c + 2 = 6.3040894096... where c is this constant (Seiffert, 1994). - Amiram Eldar, Jan 15 2022

Equals Sum_{k>=1} F(k)/(k*3^k), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Jul 02 2023

Equals 1/Product_{p prime} (1 - Kronecker(5,p)/p), where Kronecker(5,p) = 0 if p = 5, 1 if p == 1 or 4 (mod 5) or -1 if p == 2 or 3 (mod 5). - Amiram Eldar, Dec 17 2023

Equals A344041/2. - Hugo Pfoertner, Oct 16 2024