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%I #17 Mar 25 2024 15:35:49
%S 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 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 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 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 B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 102.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>
%F Equals 2*sqrt(2*Pi)/(3*Gamma(3/4)^2).
%F From _Peter Bala_, Mar 24 2024: (Start)
%F 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 For example, taking k = 0 and k = 1 yields
%F 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 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 Equals (2/3) * 1/A076390. (End)
%e 1.11283578889876424837523964373206241199199...
%t L = Pi^(3/2)/(Sqrt[2]*Gamma[3/4]^2); RealDigits[4*L/(3*Pi), 10, 103] // First
%Y Cf. A062539 (L), A076390, A085565, A225119 (L/3).
%K nonn,cons,easy
%O 1,4
%A _Jean-François Alcover_, Jun 03 2014
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