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 A085565 Decimal expansion of lemniscate constant A. 9
 1, 3, 1, 1, 0, 2, 8, 7, 7, 7, 1, 4, 6, 0, 5, 9, 9, 0, 5, 2, 3, 2, 4, 1, 9, 7, 9, 4, 9, 4, 5, 5, 5, 9, 7, 0, 6, 8, 4, 1, 3, 7, 7, 4, 7, 5, 7, 1, 5, 8, 1, 1, 5, 8, 1, 4, 0, 8, 4, 1, 0, 8, 5, 1, 9, 0, 0, 3, 9, 5, 2, 9, 3, 5, 3, 5, 2, 0, 7, 1, 2, 5, 1, 1, 5, 1, 4, 7, 7, 6, 6, 4, 8, 0, 7, 1, 4, 5, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This number is transcendental by a result of Schneider on elliptic integrals. - Benoit Cloitre, Jan 08 2006 REFERENCES Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934). Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937). LINKS G. C. Greubel, Table of n, a(n) for n = 1..5000 S. Khrushchev, Orthogonal polynomials and continued fractions from Euler’s point of view, Encyclopedia of Mathematics and its Applications 122. J. Todd, The lemniscate constants, Comm. ACM, 18 (1975), 14-19; 18 (1975), 462. Eric Weisstein's World of Mathematics, Lemniscate Constant FORMULA Equals (1/4)*(2*Pi)^(-1/2)*GAMMA(1/4)^2. Integral_{1}^{infty}dx/sqrt(4x^3-4x)=Gamma(1/4)^2/4/sqrt(2*Pi)= 1.31102877714605990523... . - Benoit Cloitre, Jan 08 2006 Equals prod(k>=0, [(4k+3)(4k+2)] / [(4k+5)(4k+4)] ) (Gauss). - Ralf Stephan, Mar 04 2008 Equals Pi/sqrt(8)/agm(1,sqrt(1/2)). Equals Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2). Prod_{m>=1} ((2*m)/(2*m+1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch. From Peter Bala, Mar 09 2015: (Start) Equals int_{x = 0..1} 1/sqrt(1 - x^4) dx. Continued fraction representations: 2/(1 + 1*3/(2 + 5*7/(2 + 9*11/(2 + ... )))) due to Euler - see Khrushchev, p. 179. Also equals 1 + 1/(2 + 2*3/(2 + 4*5/(2 + 6*7/(2 + ... )))). (End) EXAMPLE 1.3110287771460599052324197949455597068413774757158115814084108519... MATHEMATICA RealDigits[ Gamma[1/4]^2/(4*Sqrt[2*Pi]), 10, 99][[1]] (* or *) RealDigits[ EllipticK[-1], 10, 99][[1]] (* Jean-François Alcover, Mar 07 2013, updated Jul 30 2016 *) PROG (PARI) gamma(1/4)^2/4/sqrt(2*Pi) (PARI) K(x)=Pi/2/agm(1, sqrt(1-x)) K(-1) \\ Charles R Greathouse IV, Aug 02 2018 (MAGMA) C := ComplexField(); [Gamma(1/4)^2/(4*Sqrt(2*Pi(C)))]; // G. C. Greubel, Nov 05 2017 CROSSREFS Cf. A076390. Sequence in context: A011354 A143119 A220419 * A216677 A196057 A058395 Adjacent sequences:  A085562 A085563 A085564 * A085566 A085567 A085568 KEYWORD nonn,cons AUTHOR N. J. A. Sloane, Jul 06 2003 STATUS approved

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Last modified December 12 00:07 EST 2018. Contains 318052 sequences. (Running on oeis4.)