cp's OEIS Frontend

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

%I A155969 #14 Aug 28 2025 00:33:23
%S A155969 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,
%T A155969 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,
%U A155969 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
%N A155969 Decimal expansion of the square of the Euler-Mascheroni constant.
%C A155969 The Pierce expansion is 3, 2144, 2463, 5226, 17239, 51372, 287963, 387316, 3226210,...
%C A155969 From _Peter Bala_, Aug 24 2025: (Start)
%C A155969 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.
%C A155969 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)
%H A155969 G. C. Greubel, <a href="/A155969/b155969.txt">Table of n, a(n) for n = 0..1000</a>
%H A155969 C. Elsner, <a href="https://doi.org/10.1090/S0002-9939-1995-1233969-4">On a sequence transformation with integral coefficients for Euler's constant</a>, Proc. Amer. Math. Soc., Vol. 123 (1995), Number 5, pp. 1537-1541.
%H A155969 Simon Plouffe <a href="https://plouffe.fr/simon/constants/gamma2.txt">100,000 digits</a>
%F A155969 Equals A001620^2.
%e A155969 0.3331779238077186743183761363552442...
%p A155969 evalf(gamma^2);
%t A155969 RealDigits[N[EulerGamma^2, 100]][[1]] (* _G. C. Greubel_, Dec 26 2016 *)
%o A155969 (PARI) Euler^2 \\ _G. C. Greubel_, Dec 26 2016
%Y A155969 Cf. A001620, A002389, A073004, A098907, A346589
%K A155969 cons,easy,nonn,changed
%O A155969 0,1
%A A155969 _R. J. Mathar_, Jan 31 2009