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A249282 Decimal expansion of K(1/4), where K is the complete elliptic integral of the first kind.

%I A249282 #25 Apr 28 2025 07:09:22
%S A249282 1,6,8,5,7,5,0,3,5,4,8,1,2,5,9,6,0,4,2,8,7,1,2,0,3,6,5,7,7,9,9,0,7,6,
%T A249282 9,8,9,5,0,0,8,0,0,8,9,4,1,4,1,0,8,9,0,4,4,1,1,9,9,4,8,2,9,7,8,9,3,4,
%U A249282 3,3,7,0,2,8,8,2,3,4,6,7,6,0,4,0,6,4,5,0,9,7,3,9,3,6,6,1,2,5,7,0,3,3
%N A249282 Decimal expansion of K(1/4), where K is the complete elliptic integral of the first kind.
%H A249282 Steven R. Finch, <a href="/A249282/a249282.pdf">Gergonne-Schwarz Surface</a>, April 12, 2013. [Cached copy, with permission of the author]
%H A249282 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>.
%F A249282 From _Paul D. Hanna_, Mar 25 2024: (Start)
%F A249282 K(1/4) = Pi/2 * Sum_{n>=0} binomial(2*n,n)^2/16^n * (1/4)^n.
%F A249282 K(1/4) = Pi/2 * sqrt( Sum_{n>=0} binomial(2*n,n)^3/16^n * (m*(1-m))^n ), where m = 1/4. (End)
%F A249282 Equals Pi/agm(1, 3) = A000796 / A084895. - _Amiram Eldar_, Apr 28 2025
%e A249282 1.685750354812596042871203657799076989500800894141089...
%p A249282 evalf(EllipticK(1/2), 120); # _Vaclav Kotesovec_, Apr 22 2015
%t A249282 RealDigits[EllipticK[1/4], 10, 102] // First
%o A249282 (PARI) ellK(1/2) \\ _Charles R Greathouse IV_, Feb 04 2025
%Y A249282 Cf. A093341 (K(1/2)), A249283 (K(3/4)), A000796, A084895.
%K A249282 nonn,cons,easy
%O A249282 1,2
%A A249282 _Jean-François Alcover_, Oct 24 2014