A002389 Decimal expansion of -log(gamma), where gamma is Euler's constant A001620. (Formerly M3740)

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

Offset

0,1

Comments

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)*log(s(n+k)). Then it appears that E(n) converges rapidly to log(gamma). For example, E(50) = -0.549539312981644822337661768802(88...) gives log(gamma) correct to 30 decimal digits. Cf. A073004. (End)

References

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Ivan Panchenko, 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, -log(gamma) to 10000 digits

Simon Plouffe, -log(gamma) to 1024 digits

Example

.549539312981644822337661768802907788330698981263...

Mathematica

RealDigits[-Log[EulerGamma], 10, 100][[1]] (* G. C. Greubel, Sep 07 2018 *)

Prog

(PARI) -log(Euler) \\ Michel Marcus, Mar 11 2013
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); -Log(EulerGamma(R)); // G. C. Greubel, Sep 07 2018

Crossrefs

Cf. A001620, A073004, A155969, A213440.

Sequence in context: A212878 A376258 A092302 * A196758 A019776 A057763

Adjacent sequences: A002386 A002387 A002388 * A002390 A002391 A002392

Keyword

nonn,cons

Author

N. J. A. Sloane

Status

approved