A222183 Decimal expansion of Sum_{k >= 0} 1/(4*k+1)^2.

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

Offset

1,3

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7.2, p. 55.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

E. D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 112.

Formula

Equals A006752/2 + A222068.

Equals -Integral_{x=0..1} log(x)/(1 - x^4) dx. - Amiram Eldar, Jul 17 2020

Equals 3F2(1/4,1/4,1;5/4,5/4;1). [Krupnikov] - R. J. Mathar, Jun 12 2024

Equals psi'(1/4)/16 (see Shamos). - Stefano Spezia, Nov 12 2024

Example

1.074833072156694421204457444958451501344... = 1 + 1/25 + 1/81 + 1/169 + 1/289 + ...

Mathematica

RealDigits[Catalan/2 + Pi^2/16, 10, 90][[1]] (* or *) RealDigits[PolyGamma[1, 1/4]/16, 10, 90]

Prog

(PARI) (8*Catalan + Pi^2)/16 \\ G. C. Greubel, Aug 23 2018
(PARI) sumnumrat(1/(4*x+1)^2, 0) \\ Charles R Greathouse IV, Jul 08 2026
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (8*Catalan(R) + Pi(R)^2)/16; // G. C. Greubel, Aug 23 2018

Crossrefs

Cf. A006752, A222068.

Cf. A013661, A111003, A214550.

Sequence in context: A195384 A394107 A021576 * A010509 A161166 A199060

Adjacent sequences: A222180 A222181 A222182 * A222184 A222185 A222186

Keyword

nonn,cons,changed

Author

Bruno Berselli, Feb 11 2013

Status

approved