A076390 Decimal expansion of lemniscate constant B.

5, 9, 9, 0, 7, 0, 1, 1, 7, 3, 6, 7, 7, 9, 6, 1, 0, 3, 7, 1, 9, 9, 6, 1, 2, 4, 6, 1, 4, 0, 1, 6, 1, 9, 3, 9, 1, 1, 3, 6, 0, 6, 3, 3, 1, 6, 0, 7, 8, 2, 5, 7, 7, 9, 1, 3, 1, 8, 3, 7, 4, 7, 6, 4, 7, 3, 2, 0, 2, 6, 0, 7, 0, 7, 1, 9, 5, 7, 8, 3, 5, 4, 1, 7, 9, 4, 2, 7, 7, 8, 2, 4, 4, 8, 9, 6, 6, 9, 4, 6, 8, 7, 9, 5, 3, 6

Offset

0,1

Comments

Also decimal expansion of AGM(1,i)/(1+i).

See A085565 for the lemniscate constant A. - Peter Bala, Oct 25 2019

Also the ratio of height to diameter of a "Mylar balloon" (see Paulsen). - Jeremy Tan, May 05 2021

References

J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1998.

Links

Table of n, a(n) for n=0..105.

W. H. Paulsen, What Is the Shape of a Mylar Balloon?, Amer. Math. Monthly 101 (10), (Dec. 1994), pp. 953-958.

J. Todd, The lemniscate constants, Comm. ACM, Vol. 18, No. 1 (1975), pp. 14-19; corrigendum, Vol. 18, No. 8 (1975), p. 462.

J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.

Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean

Eric Weisstein's World of Mathematics, Lemniscate Constant

Wikipedia, Mylar balloon

Wolfram Research, Arithmetic-Geometric Mean.

Formula

Equals (2*Pi)^(-1/2)*GAMMA(3/4)^2.

Equals ee/sqrt(2)-1/2*sqrt(2*ee^2-Pi) where ee = EllipticE(1/2), or also prod_{m>=1} ((2*m)/(2*m-1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch.

Equals sqrt(2) * Pi^(3/2) / GAMMA(1/4)^2. - Vaclav Kotesovec, Oct 03 2019

From Peter Bala, Oct 25 2019: (Start)

Equals 1 - 1/3 - 1/(3*7) - (1*3)/(3*7*11) - (1*3*5)/(3*7*11*15) - ... = hypergeom([-1/2,1],[3/4],1/2) by Gauss’s second summation theorem.

Equivalently, define a sequence of rational numbers r(n) recursively by r(n) = (2*n - 3)/(4*n - 1)*r(n-1) with r(0) = 1. Then the constant equals Sum_{n >= 0} r(n) = 1 - 1/3 - 1/21 - 1/77 - 1/231 - 1/627 - 3/4807 - 1/3933 - 13/121923 - 13/284487 - 17/853461 - .... The partial sum of the series to 100 terms gives the constant correct to 32 decimal places.

Equals (1/3) + (1*3)/(3*7) + (1*3*5)/(3*7*11) + ... = (1/3) * hypergeom ([3/2,1],[7/4],1/2). (End)

Equals (1/2) * A053004. - Amiram Eldar, Aug 26 2020

Equals (2/3) * 1/A243340. - Peter Bala, Mar 25 2024

Example

0.599070117367796103719961246140161939113606331607825779131837476473202607...
AGM(1,i) = 0.59907011736779610371... + 0.59907011736779610371...*i

Mathematica

RealDigits[ Chop[ N[ ArithmeticGeometricMean[1, I]/(1 + I), 111]]] [[1]]
RealDigits[N[Pi/(4 EllipticK[-1]), 106]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

Prog

(PARI) real(agm(1, I)/(1+I)) \\ Charles R Greathouse IV, Mar 03 2016
(PARI) (2*Pi)^(-1/2)*gamma(3/4)^2 \\ Michel Marcus, Nov 10 2017

Crossrefs

Cf. A053004, A076391, A076392, A085565, A243340.

Sequence in context: A123600 A063623 A085566 * A321632 A147818 A345739

Adjacent sequences: A076387 A076388 A076389 * A076391 A076392 A076393

Keyword

nonn,cons

Author

Robert G. Wilson v, Oct 09 2002

Extensions

Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar

Status

approved