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

Author

Bruno Berselli, Feb 11 2013

Keywords

• nonn
• cons

Examples

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

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.

Crossrefs

Cf. A006752, A222068.

Cf. A013661, A111003, A214550.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); (8*Catalan(R) + Pi(R)^2)/16; // G. C. Greubel, Aug 23 2018

Mathematica

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

PARI

(8*Catalan + Pi^2)/16 \\ G. C. Greubel, Aug 23 2018

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