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%I A086466 #60 Oct 16 2024 09:22:41
%S A086466 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,
%T A086466 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,
%U A086466 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
%N A086466 Decimal expansion of 2*sqrt(5)/5 arccsch(2).
%C A086466 Equals the value of the Dirichlet L-series of the non-principal character modulo 5 (A080891) at s=1. - _Jianing Song_, Nov 16 2019
%D A086466 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.2, p. 7.
%H A086466 R. J. Mathar, <a href="https://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 22 for L(m=5,r=3,s=1).
%H A086466 H.-J. Seiffert, <a href="https://fq.math.ca/Scanned/32-4/elementary32-4.pdf">Problem B-771</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 32, No. 4 (1994), p. 374; <a href="https://www.fq.math.ca/Scanned/33-5/elementary33-5.pdf">More Sums</a>, Solution to Problem B-771 by Don Redmond, ibid., Vol. 33, No. 5 (1995), pp. 470-471.
%H A086466 Renzo Sprugnoli, <a href="https://www.emis.de/journals/INTEGERS/papers/g27/g27.Abstract.html">Sums of reciprocals of the central binomial coefficients</a>, INTEGERS 6 (2006) #A27
%H A086466 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CentralBinomialCoefficient.html">Central Binomial Coefficient</a>.
%F A086466 Equals Sum_{k>=1} (-1)^(k-1)/(k*binomial(2*k,k)).
%F A086466 Equals A010532 * A002390 / 10. - _R. J. Mathar_, Jul 26 2010
%F A086466 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
%F A086466 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
%F A086466 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
%F A086466 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
%F A086466 Equals Sum_{k>=1} F(k)/(k*3^k), where F(k) is the k-th Fibonacci number (A000045). - _Amiram Eldar_, Jul 02 2023
%F A086466 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
%F A086466 Equals A344041/2. - _Hugo Pfoertner_, Oct 16 2024
%e A086466 0.43040894096400403888943323295060542542457...
%t A086466 2*Log[GoldenRatio]/Sqrt[5] // RealDigits[#, 10, 102]& // First (* _Jean-François Alcover_, Apr 18 2014 *)
%o A086466 (PARI) 2*log((1+sqrt(5))/2)/sqrt(5) \\ _Stefano Spezia_, Oct 15 2024
%Y A086466 Cf. A086465, A086467, A086468.
%Y A086466 Cf. A000032, A000045, A080891, A328717 (s=2), A344041.
%K A086466 nonn,cons
%O A086466 0,1
%A A086466 _Eric W. Weisstein_, Jul 21 2003

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