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 A243340 #21 Feb 16 2025 08:33:22 %S A243340 1,1,1,2,8,3,5,7,8,8,8,9,8,7,6,4,2,4,8,3,7,5,2,3,9,6,4,3,7,3,2,0,6,2, %T A243340 4,1,1,9,9,1,9,9,0,6,8,4,6,5,3,7,9,6,0,0,3,2,6,6,4,3,6,4,9,3,4,7,1,5, %U A243340 7,5,9,9,0,2,7,9,3,6,8,5,4,9,1,5,9,5,8,8,2,1,3,8,0,1,7,0,0,4,3,2,1,7,2,0,9 %N A243340 Decimal expansion of 4*L/(3*Pi), a constant related to the asymptotic evaluation of the number of primes of the form a^2+b^4, where L is Gauss' lemniscate constant. %D A243340 B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25. %D A243340 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102. %H A243340 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>. %F A243340 Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2). %F A243340 From _Peter Bala_, Mar 24 2024: (Start) %F A243340 An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 3 for k >= 0. %F A243340 For example, taking k = 0 and k = 1 yields %F A243340 4*L/(3*Pi) = 1 + 1/(6 + (5*7)/(6 + (9*11)/(6 + (13*15)/(6 + ... + (4*n + 1)*(4*n + 3)/(6 + ... ))))) and %F A243340 4*L/(3*Pi) = 8/(7 + (1*3)/(14 + (5*7)/(14 + (9*11)/(14 + (13*15)/(14 + ... + (4*n + 1)*(4*n + 3)/(14 + ... )))))). %F A243340 Equals (2/3) * 1/A076390. (End) %e A243340 1.11283578889876424837523964373206241199199... %t A243340 L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First %o A243340 (PARI) 2*sqrt(2*Pi)/(3*gamma(3/4)^2) \\ _Stefano Spezia_, Nov 27 2024 %Y A243340 Cf. A062539 (L), A076390, A085565, A225119 (L/3). %K A243340 nonn,cons,easy %O A243340 1,4 %A A243340 _Jean-François Alcover_, Jun 03 2014