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

Author

R. J. Mathar, Jan 31 2009

Keywords

• cons
• easy
• nonn
• changed

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)

Examples

			0.3331779238077186743183761363552442...
		

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

Crossrefs

Cf. A001620, A002389, A073004, A098907, A346589

Programs

Maple

evalf(gamma^2);

Mathematica

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

PARI

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

Formula

Equals A001620^2.