A155969 Decimal expansion of the square of the Euler-Mascheroni constant.

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

Offset

0,1

Comments

The Pierce expansion is 3, 2144, 2463, 5226, 17239, 51372, 287963, 387316, 3226210,...

From Peter Bala, Aug 24 2025: (Start)

By definition, the Euler-Mascheroni constant gamma = lim_{n -> oo} s(n), where s(n) = Sum_{k = 1..n} 1/k - log(n). The convergence is slow. For example, s(50) = 0.5(87...) is only correct to 1 decimal digit. Let S(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k). Elsner shows that S(n) converges to gamma much more rapidly. For example, S(50) = 0.57721566490153286060651209008(02...) gives gamma correct to 29 decimal digits.

Define E(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*binomial(n+k, k)*s(n+k)^2. Then it appears that E(n) converges rapidly to gamma^2. For example, E(50) = 0.33317792380771867431837613635524(22...) gives gamma^2 correct to 32 decimal digits. (End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

C. Elsner, On a sequence transformation with integral coefficients for Euler's constant, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.

Simon Plouffe 100,000 digits

Formula

Equals A001620^2.

Example

0.3331779238077186743183761363552442...

Maple

evalf(gamma^2);

Mathematica

RealDigits[N[EulerGamma^2, 100]][[1]] (* G. C. Greubel, Dec 26 2016 *)

Prog

(PARI) Euler^2 \\ G. C. Greubel, Dec 26 2016

Crossrefs

Cf. A001620, A002389, A073004, A098907, A346589

Sequence in context: A226645 A243095 A304586 * A076237 A201432 A128210

Adjacent sequences: A155966 A155967 A155968 * A155970 A155971 A155972

Keyword

cons,easy,nonn

Author

R. J. Mathar, Jan 31 2009

Status

approved