This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A222882 #36 Jun 28 2023 08:20:03
%S A222882 2,2,5,4,9,2,2,4,6,2,8,8,8,2,6,4,7,6,6,2,6,8,1,8,4,7,5,9,5,2,8,7,2,3,
%T A222882 5,5,7,8,7,1,6,6,1,5,9,8,6,0,5,3,5,1,8,8,9,1,3,8,3,1,1,6,1,8,8,5,9,1,
%U A222882 7,2,9,2,8,9,5,9,7,1,3,9,3,4,1,0,5,8
%N A222882 Decimal expansion of Sierpiński's second constant, K2 = lim_{n->oo} ((1/n) * (Sum_{i=1..n} A004018(i^2)) - 4/Pi * log(n)).
%C A222882 Sierpiński introduced three constants in his 1908 doctoral thesis. The first, K, is very well known, bears his name and its decimal expansion is given in A062089. However, the second and third of these constants appear to have been largely forgotten. This sequence gives the decimal expansion of the second one, K2, and A222883 gives the decimal expansion of the third , K3. The formula given below show that K2 is related to several other, naturally occurring constants.
%D A222882 Steven R. Finch, Mathematical Constants, Encyclopaedia of Mathematics and its Applications, Cambridge University Press (2003), p.123. Corrigenda in the link below.
%H A222882 G. C. Greubel, <a href="/A222882/b222882.txt">Table of n, a(n) for n = 1..10000</a>
%H A222882 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, (June 2012), pp. 15-16.
%H A222882 A. Schinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa2112.pdf">Wacław Sierpiński’s papers on the theory of numbers</a>, Acta Arithmetica XXI, (1972), pp. 7-13. Corrigenda in "Dzieje Matematyki Polskiej" (Wrocław 2012), p.228 (in Polish).
%F A222882 K2 = 4 / Pi * (eulergamma + K / Pi - 12 / Pi^2 * zeta'(2) + log(2) / 3 -1), where K is Sierpiński's first constant (A062089) and eulergamma is the Euler-Mascheroni constant (A001620).
%F A222882 K2 = 4 * (12 * log(Gamma(3/4)) - 9*log(Pi) + 72*log(A) - 5*log(2) + 3 * eulergamma - 3) / (3 * Pi), where A is the Glaisher-Kinkelin constant (A074962).
%F A222882 K2 = 4 * (12 * log(Gamma(3/4)) + log(A^72 * e^(3*eulergamma - 3) / (32 * Pi^9))) / (3 * Pi).
%F A222882 K2 = 4 / Pi * (log(e^(3*eulergamma - 1) / (2^(2/3) * G^2)) - 12 / Pi^2 * zeta'(2)), where G is Gauss’ AGM constant (A014549).
%F A222882 K2 = 4 / Pi * (log(Pi^2 * e^(3*eulergamma - 1) / (2^(2/3) * L^2)) - 12 / Pi^2 * zeta'(2)), where L is Gauss’ lemniscate constant (A062539).
%e A222882 K2 = 2.25492246288826476626818475952872355787166159860535188913831...
%t A222882 Take[Flatten[RealDigits[N[4(12 Log[Gamma[3/4]]-9 Log[Pi]+72 Log[Glaisher]-5 Log[2]+3 EulerGamma-3)/(3 Pi),100]]],86]
%o A222882 (PARI) 4/Pi*(log(exp(3*Euler-1)/(2^(2/3)/agm(sqrt(2),1)^2)) - 12/Pi^2*zeta'(2)) \\ _Charles R Greathouse IV_, Dec 12 2013
%Y A222882 Cf. A001620, A004018, A014549, A062089, A062539, A074962, A222883.
%K A222882 nonn,cons
%O A222882 1,1
%A A222882 _Ant King_, Mar 11 2013
%E A222882 Minor edits by _Vaclav Kotesovec_, Nov 14 2014