A338106 Decimal expansion of Sum_{m>1, n>1} 1/(m^2*n^2-1).

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

Offset

0,1

Comments

For p>1, q>1 in R, Sum_{m >1, n>1} 1/(m^p*n^q-1) = Sum_{k>0} (zeta(k*p) - 1) * (zeta(k*q) - 1) [Proof in References]. This sequence corresponds to p = q = 2.

Double inequality: Sum_{m>1, n>1} 1/(m^2*n^2+1) = A338107 = 0.409... < Sum_{m>1, n>1} 1/(m^2*n^2) = (zeta(2)-1)^2 = 0.415... < Sum_{m>1, n>1} 1/(m^2*n^2-1) = this constant = 0.423...

References

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.25, p. 277.

Links

Table of n, a(n) for n=0..87.

Formula

Equals Sum_{k>0} (zeta(2*k) - 1)^2.

Equals -3/4 + Sum_{k>=2} (1/2 - Pi*cot(Pi/k)/(2*k)). - Vaclav Kotesovec, Oct 14 2020

Example

0.4230355257613131597420971016391038628995464... (with help of Amiram Eldar).

Mathematica

RealDigits[Sum[(Zeta[2*k] - 1)^2, {k, 1, 100}], 10, 90][[1]] (* Amiram Eldar, Oct 10 2020 *)

Prog

(PARI) sumpos(k=1, (zeta(2*k) - 1)^2) \\ Michel Marcus, Oct 10 2020

Crossrefs

Cf. A098198, A333972, A338107.

Sequence in context: A359060 A134977 A199081 * A232462 A266141 A266147

Adjacent sequences: A338103 A338104 A338105 * A338107 A338108 A338109

Keyword

nonn,cons

Author

Bernard Schott, Oct 10 2020

Status

approved