A233091 - OEIS

A233091

Decimal expansion of Sum_{i>=0} 1/(2*i+1)^3.

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

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OFFSET

1,3

COMMENTS

This constant is irrational. - Charles R Greathouse IV, Feb 03 2025

REFERENCES

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

LINKS

Table of n, a(n) for n=1..86.

J. M. Borwein, I.J. Zucker, and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, Volume 15, Issue 3 (April 2008), pp. 377-405, see p. 17 c(3); alternative link.

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.

Seán Mark Stewart, Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers, Tatra Mountains Mathematical Publications, Vol. 77, No. 1 (2020), pp. 73-98; alternative link. See p. 75, eq. (4).

FORMULA

Equals 7*zeta(3)/8.

Also equals -(1/16)*PolyGamma(2, 1/2). - Jean-François Alcover, Dec 18 2013

Equals Integral_{x=0..Pi/2} x * log(tan(x)) dx. - Amiram Eldar, Jun 29 2020

Equals Integral_{x=0..1} arcsin(x)*arccos(x)/x dx. - Amiram Eldar, Aug 03 2020

Equals Sum_{k>=1} H(k)^2/(k*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Stewart, 2020, p. 79, eq. (12)). - Amiram Eldar, Feb 03 2026

EXAMPLE

1.0517997902646449997247708913225187419193630057979365215682376109241...

MATHEMATICA

RealDigits[7 Zeta[3]/8, 10, 90][[1]]

PROG

(PARI) 7*zeta(3)/8 \\ Stefano Spezia, Oct 31 2024

CROSSREFS

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233090 (5*zeta(3)/8).

Cf. A153071 (Sum_{i >= 0} (-1)^i/(2*i+1)^3).

Cf. A251809 (Sum_{i >= 0} (-1)^floor(i/2)/(2*i+1)^3).

Cf. A001008, A002805, A016755.

Sequence in context: A342921 A342417 A380862 * A286941 A332459 A051854