A249282 - OEIS

%I #20 Feb 04 2025 22:42:35

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

%H Steven R. Finch, <a href="/A249282/a249282.pdf">Gergonne-Schwarz Surface</a>, April 12, 2013. [Cached copy, with permission of the author]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>

%F From _Paul D. Hanna_, Mar 25 2024: (Start)

%F K(1/4) = Pi/2 * Sum_{n>=0} binomial(2*n,n)^2/16^n * (1/4)^n.

%F 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)

%e 1.685750354812596042871203657799076989500800894141089...

%p evalf(EllipticK(1/2), 120); # _Vaclav Kotesovec_, Apr 22 2015

%t RealDigits[EllipticK[1/4], 10, 102] // First

%o (PARI) ellK(1/2) \\ _Charles R Greathouse IV_, Feb 04 2025

%Y Cf. A093341 (K(1/2)), A249283 (K(3/4)).

%K nonn,cons,easy,changed

%O 1,2

%A _Jean-François Alcover_, Oct 24 2014