A214552 -id:A214552 - cp's OEIS frontend

A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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

Offset: 0

N. J. A. Sloane, Jul 31 2003

This number is L(2, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3, A102283. - Stuart Clary, Dec 17 2008
Equals 1/1^2 -1/2^2 +1/4^2 -1/5^2 +1/7^2 -1/8^2 +1/10^2 -1/11^2 +-... . This can be split as (1/1^2 -1/5^2 +1/7^2 -1/11^2 +-...) - (1/2^2 -1/4^2 +1/8^2 -1/10^2..) = (g(1)-g(5)) - (g(2)-g(4)). The first of these two series is A214552 and the second series is 1/(2^2)*(1-1/2^2 +1/4^2-1/5^2+-...), namely a quarter of the original series. Therefore 5/4 of this value here equals A214552. - R. J. Mathar, Jul 20 2012
Calegari, Dimitrov, & Tang prove that this number is irrational. - Charles R Greathouse IV, Aug 29 2024

			0.781302412896486296867...

L. Fejes Toth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

D. H. Bailey, J. M. Borwein, and R. E. Crandall, Integrals of the Ising class, J. Phys. A 39 (2006) 12271, variable C_3.
Frank Calegari, Vesselin Dimitrov, and Yunqing Tang, The linear independence of 1, ζ(2), and L(2,χ₋₃), arXiv:2408.15403 [math.NT], 2024.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 98.
Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, L(m=3,r=2,s=2).
Lorenz Milla, World record computation of Dirichlet L(-3,2) series (1,000,000,000,000 digits)
Kh. Hessami Pilehrood and T. Hessami Pilehrood, Bivariate identities for values of the Hurwitz zeta function and supercongruences, El. J. Combin. 18 (2) (2012), Article P35, value of K after Theorem 4.

Cf. A086722-A086731, A102283, A214549 (principal character), A214552.

Cf. A153066, A153067, A153068.

nmax = 1000; First[ RealDigits[(Zeta[2, 1/3] - Zeta[2, 2/3])/9, 10, nmax] ] (* Stuart Clary, Dec 17 2008 *)

zetahurwitz(2,1/3)/9 - zetahurwitz(2,2/3)/9 \\ Charles R Greathouse IV, Jan 30 2018

From Jean-François Alcover, Jul 17 2014, updated Jan 23 2015: (Start)
Equals Sum_{n>=1} jacobi(-3, n+3)/n^2.
Equals (8/15)*4F3(1/2,1,1,2; 5/4,3/2,7/4; 3/4), where 4F3 is the generalized hypergeometric function.
Equals 4*Pi*log(3)/(3*sqrt(3)) - 4*Integral_{0..1} log(x+1)/(x^2-x+1) dx. (End)
Equals Product_{p prime} (1 - Kronecker(-3, p)/p^2)^(-1) = Product_{p prime != 3} (1 + (-1)^(p mod 3)/p^2)^(-1). - Amiram Eldar, Nov 06 2023

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

Original entry on oeis.org

0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 0

Paul Curtz, Jan 26 2008

Dirichlet series for the non-principal character modulo 6: L(s,chi) = Sum_{n>=1} a(n)/n^s. For example L(1,chi) = A093766, L(2,chi) = A214552, and L(3,chi) = Pi^3/(18*sqrt(3)). See Jolley eq. (314) and arXiv:1008.2547 L(m=6,r=2,s). - R. J. Mathar, Jul 31 2010

			G.f. = x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=6, Chi_2(n).
L. B. W. Jolley, Summation of Series, Dover (1961).

R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A093766, A120325, A131719, A131720, A131735, A131736, A214552.

&cat[[0, 1, 0, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A134667:=n->[0, 1, 0, 0, 0, -1][(n mod 6)+1]: seq(A134667(n), n=0..100);
# Wesley Ivan Hurt, Jun 20 2016

a[ n_] := JacobiSymbol[-12, n]; (* Michael Somos, Apr 24 2014 *)
a[ n_] := {1, 0, 0, 0, -1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Apr 24 2014 *)
PadRight[{},120,{0,1,0,0,0,-1}] (* Harvey P. Dale, Aug 01 2021 *)

{a(n) = [0, 1, 0, 0, 0, -1][n%6+1]}; /* Michael Somos, Feb 10 2008 */

{a(n) = kronecker(-12, n)}; /* Michael Somos, Feb 10 2008 */

{a(n) = if( n < 0, -a(-n), if( n<1, 0, direuler(p=2, n, 1 / (1 - kronecker(-12, p) * X))[n]))}; /* Michael Somos, Aug 11 2009 */

Euler transform of length 6 sequence [0, 0, 0, -1, 0, 1]. - Michael Somos, Feb 10 2008
G.f.: x * (1 - x^4) / (1 - x^6) = x*(1+x^2) / (1 + x^2 + x^4) = x*(1+x^2) / ( (1+x+x^2)*(x^2-x+1) ).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3)) where f(u, v, w) = w * (2 + v - u^2 - 2*v^2) - 2 * u * v. - Michael Somos, Aug 11 2009
a(n) is multiplicative with a(p^e) = 0^e if p = 2 or p = 3, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Aug 11 2009
a(-n) = -a(n). a(n+6) = a(n). a(2*n) = a(3*n) = 0.
sqrt(3)*a(n) = sin(Pi*n/3) + sin(2*Pi*n/3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: 2*sin(sqrt(3)*x/2)*cosh(x/2)/sqrt(3). - Ilya Gutkovskiy, Jun 21 2016

A086722 Decimal expansion of g(1)+g(2)-g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jul 31 2003

By summing over g(1)-g(5) and g(2)-g(4) separately we obtain A214552 for the first difference and a quarter of A086724 for the second difference. - R. J. Mathar, Jul 20 2012

2/3 times this constant equals A086724 [Bailey, Borwein and Crandall, 2006] - R. J. Mathar, Jul 20 2012

			1.1719536193447294453... = A214552 + A086724/4 = 1/1^2 +1/2^2 -1/4^2 -1/5^2 +1/7^2 +1/8^2 -1/10^2 -1/11^2 ++--....

L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd. ed., Springer-Verlag, Berlin, Heidelberg 1972; see p. 213.

David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall, Integrals of the Ising class, Journal of Physics A: Mathematical and General, Vol. 39, No. 40 (2006), 12271; alternative link.

Cf. A086723-A086731, A214552.

g[k_] := PolyGamma[1, k/6]/36; RealDigits[g[1] + g[2] - g[4] - g[5], 10, 99] // First (* Jean-François Alcover, Feb 12 2013 *)

Equals -Integral_{x=0..1} log(x)/(x^2-x+1) dx. - Jean-François Alcover, Aug 29 2014

Equals Integral_{x>=0} x/(exp(x) + exp(-x) - 1) dx. - Amiram Eldar, May 22 2023

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A086724 Decimal expansion of L(2, chi3) = g(1)-g(2)+g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

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A086722 Decimal expansion of g(1)+g(2)-g(4)-g(5), where g(k) = Sum_{m>=0} (1/(6*m+k)^2).

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