A243340 -id:A243340 - cp's OEIS frontend

A085565 Decimal expansion of lemniscate constant A.

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

Offset: 1

N. J. A. Sloane, Jul 06 2003

This number is transcendental by a result of Schneider on elliptic integrals. - Benoit Cloitre, Jan 08 2006
The two lemniscate constants are A = Integral_{x = 0..1} 1/sqrt(1 - x^4) dx and B = Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx. See A076390. - Peter Bala, Oct 25 2019
Also the ratio of generating curve length to diameter of a "Mylar balloon" (see Paulsen). - Jeremy Tan, May 05 2021

			1.3110287771460599052324197949455597068413774757158115814084108519...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934).
Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937).

G. C. Greubel, Table of n, a(n) for n = 1..5000
John Maxwell Campbell, WZ proofs for lemniscate-like constant evaluations, Integers 21 (2021), Article A107, 15.
S. Khrushchev, Orthogonal polynomials and continued fractions from Euler’s point of view, Encyclopedia of Mathematics and its Applications 122.
Rensley Meulens, A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation, AIP Advances (2022) Vol. 12, 015308.
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, 18 (1975), 14-19; 18 (1975), 462.
J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Wikipedia, Mylar balloon.
Index entries for transcendental numbers.

Cf. A076390, A225119, A243340.

C := ComplexField(); [Gamma(1/4)^2/(4*Sqrt(2*Pi(C)))]; // G. C. Greubel, Nov 05 2017

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 *)

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

PARI

ellK(I) \\ Charles R Greathouse IV, Feb 04 2025

Formula

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

Equals Integral_{x>=1}dx/sqrt(4x^3-4x). - Benoit Cloitre, Jan 08 2006

Equals Product_(k>=0, [(4k+3)(4k+4)] / [(4k+5)(4k+2)] ) (Gauss). - Ralf Stephan, Mar 04 2008 [corrected by Vaclav Kotesovec, May 01 2020]

Equals Pi/sqrt(8)/agm(1,sqrt(1/2)).

Equals Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2).

Product_{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 Integral_{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)

From Peter Bala, Oct 25 2019: (Start)

Equals 1 + 1/5 + (1*3)/(5*9) + (1*3*5)/(5*9*13) + ... = hypergeom([1/2,1],[5/4],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 - 3)*r(n-1) with r(1) = 1. Then the constant equals Sum_{n >= 1} r(n) = 1 + 1/5 + 1/15 + 1/39 + 7/663 + 1/221 + 11/5525 + 11/12325 + 1/2465 + .... The partial sum of the series to 100 terms gives 32 correct decimal digits for the constant.

Equals (1*3)/(1*5) + (1*3*5)/(1*5*9) + (1*3*5*7)/(1*5*9*13) + ... = (3/5) * hypergeom([5/2,1],[9/4],1/2). (End)

Equals (3/2)*A225119. - Peter Bala, Oct 27 2019

Equals Integral_{x=0..Pi/2} 1/sqrt(1 + cos(x)^2) dx = Integral_{x=0..Pi/2} 1/sqrt(1 + sin(x)^2) dx. - Amiram Eldar, Aug 09 2020

From Peter Bala, Mar 24 2024: (Start)

An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 1 for k >= 0.

For example, taking k = 0 and k = 1 yields

A = 2/(1 + (1*3)/(2 + (5*7)/(2 + (9*11)/(2 + (13*15)/(2 + ... + (4*n + 1)*(4*n + 3)/(2 + ... )))))) and

A = (1/4)*(5 + (1*3)/(10 + (5*7)/(10 + (9*11)/(10 + (13*15)/(10 + ... + (4*n + 1)*(4*n + 3)/(10 + ... )))))). (End)

A076390 Decimal expansion of lemniscate constant B.

Original entry on oeis.org

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

Robert G. Wilson v, Oct 09 2002

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

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

J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1998.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.

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.

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

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

real(agm(1,I)/(1+I)) \\ Charles R Greathouse IV, Mar 03 2016

(2*Pi)^(-1/2)*gamma(3/4)^2 \\ Michel Marcus, Nov 10 2017

Pi/4/ellK(I) \\ Charles R Greathouse IV, Feb 04 2025

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 Product_{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
Equals Product_{n>=1} exp(((-1)^n*beta(n))/n), where beta(n) is the Dirichlet beta function. - Antonio Graciá Llorente, Oct 16 2024
Equals Integral_{x=0..1} x^2/sqrt(1 - x^4) dx = sqrt(Pi)*Gamma(7/4)/(3*Gamma(5/4)) (see Finch). - Stefano Spezia, Dec 15 2024

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

A247858 Decimal expansion of the value of the continued fraction [0; 2, 5, 17, 17, 37, 41, 97, 97, ...], generated with primes of the form a^2 + b^4.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 25 2014

			1/(2 + 1/(5 + 1/(17 + 1/(17 + 1/(37 + 1/(41 + 1/(97 + 1/(97 + ...))))))))
0.45502481649017002236905280827974482410575554890507644...

John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS February 18, 1997 94 (4) 1054-1058.
Wikipedia, Friedlander-Iwaniec theorem
Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 [math.NT], 2010, pp. 8-9.

Cf. A028916, A243340, A247857.

max = 1000; r = Reap[Do[n = a^2 + b^4; If[n <= max && PrimeQ[n], Sow[n]], {a, Sqrt[max]}, {b, max^(1/4)}]][[2, 1]]; u = Union[r, SameTest -> (False&)] ; RealDigits[FromContinuedFraction[Join[{0}, u]], 10, 103] // First

cp's OEIS Frontend

A085565 Decimal expansion of lemniscate constant A.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A076390 Decimal expansion of lemniscate constant B.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A247858 Decimal expansion of the value of the continued fraction [0; 2, 5, 17, 17, 37, 41, 97, 97, ...], generated with primes of the form a^2 + b^4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica