A256780 Decimal expansion of the generalized Euler constant gamma(2,5).

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

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), p. 134.

Formula

Equals EulerGamma/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2).

Equals Sum_{n>=0} (1/(5n+2) - 2/5*arctanh(5/(10n+9))).

Equals -(psi(2/5) + log(5))/5 = (A200136 - A016628)/5. - Amiram Eldar, Jan 07 2024

Example

0.190389326430203154225983229764268163260151948448458487...

Mathematica

RealDigits[-Log[5]/5 - PolyGamma[2/5]/5, 10, 105] // First

Prog

(PARI) Euler/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2) \\ Michel Marcus, Apr 10 2015
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 - 2/Sqrt(5)) + Log(5)/20 - Sqrt(5)/10*Log((1 + Sqrt(5))/2); // G. C. Greubel, Aug 28 2018

Crossrefs

Cf. A001620 (EulerGamma), A016628, A200136, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Sequence in context: A211884 A154464 A202344 * A094452 A154705 A197389

Adjacent sequences: A256777 A256778 A256779 * A256781 A256782 A256783

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Apr 10 2015

Status

approved