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A085565 Decimal expansion of lemniscate constant A. 10
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

Index entries for transcendental numbers

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<i> := 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 October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)