A086466 - OEIS

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

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OFFSET

0,1

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

REFERENCES

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

LINKS

Table of n, a(n) for n=0..101.

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

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

EXAMPLE

0.43040894096400403888943323295060542542457...

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A086465, A086467, A086468.

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

Sequence in context: A296228 A267410 A016697 * A330392 A204694 A304030

Adjacent sequences: A086463 A086464 A086465 * A086467 A086468 A086469