A175573 -id:A175573 - cp's OEIS frontend

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A247217 Decimal expansion of theta_3(0, exp(-2*Pi)), where theta_3 is the 3rd Jacobi theta function.

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

Offset: 1

Jean-François Alcover, Nov 26 2014

			1.0037348854877390910476795950669538662079943324445194...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Jacobi Theta Functions
Wikipedia, Theta function

Cf. A000122, A175573 (theta_3(0, exp(-Pi))), A273081, A273082, A273083, A273084.

Cf. A273087, A273086.

C := ComplexField(); (4*Pi(C)*Sqrt(2) + 6*Pi(C))^(1/4)/(2*Gamma(3/4)); // G. C. Greubel, Jan 07 2018

Mathematica

RealDigits[(4*Pi*Sqrt[2] + 6*Pi)^(1/4)/(2*Gamma[3/4]), 10, 102] // First

PARI

(4*Pi*sqrt(2) + 6*Pi)^(1/4)/(2*gamma(3/4)) \\ Michel Marcus, Nov 26 2014

Formula

Equals (4*Pi*sqrt(2) + 6*Pi)^(1/4)/(2*Gamma(3/4)).

Equals Sum_{n=-infinity..infinity} exp(-2*Pi*n^2).

A273081 Decimal expansion of theta_3(0, exp(-3*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Offset: 1

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Author

Vaclav Kotesovec, May 14 2016

Keywords

nonn
cons

Examples

1.0001613990351406940215020703893995738875083912423752893728...

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273082, A273083, A273084.

Programs

Magma

C := ComplexField(); (1+2/Sqrt(3))^(1/4)*Pi(C)^(1/4)/(3^(1/4) *Gamma(3/4)) // G. C. Greubel, Jan 07 2018

Maple

evalf((1 + 2/sqrt(3))^(1/4) * Pi^(1/4) / (3^(1/4) * GAMMA(3/4)), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-3*Pi]], 10, 105][[1]] RealDigits[(1 + 2/Sqrt[3])^(1/4) * Pi^(1/4) / (3^(1/4) * Gamma[3/4]), 10, 105][[1]]

PARI

sqrtn((2/sqrt(3)+1)*Pi/3,4)/gamma(3/4) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals (1 + 2/sqrt(3))^(1/4) * Pi^(1/4) / (3^(1/4) * Gamma(3/4)).

A273082 Decimal expansion of theta_3(0, exp(-4*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Offset: 1

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Author

Vaclav Kotesovec, May 14 2016

Keywords

nonn
cons

Examples

1.00000697468471241799127935745572277338608481181934395967...

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273081, A273083, A273084.

Programs

Magma

C := ComplexField(); ((2+2^(3/4))/4)*Pi(C)^(1/4)/Gamma(3/4) // G. C. Greubel, Jan 07 2018

Maple

evalf((2+2^(3/4))/4*Pi^(1/4)/GAMMA(3/4), 120); evalf((1 + 2^(1/4)) * GAMMA(1/4) / (2^(7/4)*Pi^(3/4)), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-4*Pi]], 10, 105][[1]] RealDigits[(2 + 2^(3/4))/4 * Pi^(1/4) / Gamma[3/4], 10, 105][[1]] RealDigits[(1 + 2^(1/4)) * Gamma[1/4] / (2^(7/4)*Pi^(3/4)), 10, 105][[1]]

PARI

(2^.25+1)*gamma(1/4)/sqrtn(128*Pi^3,4) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals (2 + 2^(3/4))/4 * Pi^(1/4) / Gamma(3/4).

Equals (1 + 2^(1/4)) * Gamma(1/4) / (2^(7/4)*Pi^(3/4)).

Equals sqrt((A247217^2 + A259149^4/A259150^2)/2). - Vaclav Kotesovec, May 17 2023

A273083 Decimal expansion of theta_3(0, exp(-5*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Offset: 1

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Author

Vaclav Kotesovec, May 14 2016

Keywords

nonn
cons

Examples

1.0000003014034550780129221506549039080802236178954948667347...

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273084.

Programs

Magma

C := ComplexField(); Sqrt(2/5 + 1/Sqrt(5))*Pi(C)^(1/4)/Gamma(3/4) // G. C. Greubel, Jan 07 2018

Maple

evalf(sqrt(2/5 + 1/sqrt(5)) * Pi^(1/4)/GAMMA(3/4), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-5*Pi]], 10, 105][[1]] RealDigits[Sqrt[2/5 + 1/Sqrt[5]] * Pi^(1/4)/Gamma[3/4], 10, 105][[1]]

PARI

th3(x)=1 + 2*suminf(n=1,x^n^2) th3(exp(-5*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals sqrt(2/5 + 1/sqrt(5)) * Pi^(1/4)/Gamma(3/4).

A273084 Decimal expansion of theta_3(0, exp(-6*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Offset: 1

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Author

Vaclav Kotesovec, May 14 2016

Keywords

nonn
cons

Examples

1.0000000130248242721598014564243302309067457325414604157511...

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273083, A273086.

Programs

Magma

C := ComplexField(); Sqrt(2+Sqrt(8+6*Sqrt(3)+4*Sqrt(6 +4*Sqrt(3))))*Pi(C)^(1/4)/(2*3^(3/8)*Gamma(3/4)) // G. C. Greubel, Jan 07 2018

Maple

evalf(sqrt(2 + sqrt(8 + 6*sqrt(3) + 4*sqrt(6 + 4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8) * GAMMA(3/4)), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-6*Pi]], 10, 105][[1]] RealDigits[Sqrt[2 + Sqrt[8 + 6*Sqrt[3] + 4*Sqrt[6 + 4*Sqrt[3]]]] * Pi^(1/4) / (2*3^(3/8) * Gamma[3/4]), 10, 105][[1]]

PARI

th3(x)=1 + 2*suminf(n=1,x^n^2) th3(exp(-6*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals sqrt(2+sqrt(8+6*sqrt(3)+4*sqrt(6+4*sqrt(3)))) * Pi^(1/4) / (2*3^(3/8)*Gamma(3/4)).

Equals sqrt((A273081^2 + A292888^4/A363018^2)/2). - Vaclav Kotesovec, May 17 2023

A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

Original entry on oeis.org

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

Offset: 1

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Author

R. J. Mathar, Jul 15 2010

Keywords

cons
easy
nonn

Comments

Entry 34 c of chapter 11 of Ramanujan's second notebook.

This constant is also the ratio T(Pi/2)/T(0), where T(Pi/2) is the exact pendulum period for an amplitude of Pi/2 and T(0) the approximate period 2*Pi*sqrt(L/g) for small angles. - Jean-François Alcover, Aug 05 2014

Examples

1.18034059901609622604533794..

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.

Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76 (2008), 1150-1154.

Crossrefs

Cf. A002161, A068465, A085565, A096427, A175573.

Programs

MATLAB

sqrt(pi)/gamma(3/4)^2 % Altug Alkan, Dec 05 2015

Maple

sqrt(Pi)/GAMMA(3/4)^2 ; evalf(%) ;

Mathematica

First@ RealDigits[N[Sqrt@ Pi/Gamma[3/4]^2, 120]] (* Michael De Vlieger, Dec 06 2015 *)

PARI

sqrt(Pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015

Formula

Equals A002161 /A068465^2.

Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).

Equals 2*sqrt(2)*K(-1)/Pi, where K is the complete elliptic integral of the first kind, K(-1) being A085565. - Jean-François Alcover, Jun 03 2014

Equals Product_{k>=1} (1-(-1)^k/(2*k)) = 3/2 * 3/4 * 7/6 * 7/8 * 11/10 * 11/12 * ... . - Richard R. Forberg, Dec 05 2015

Reciprocal of A096427. Equals ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2, a rapidly converging series. For example, summing from n = -5 to n = 5 gives the constant correct to 49 decimal places. - Peter Bala, Mar 06 2019

Equals Sum_{k>=0} binomial(2*k,k)^2/2^(5*k). - Amiram Eldar, Aug 26 2020

Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - Peter Bala, Mar 04 2022

Equals A175573^2. - Amiram Eldar, Jul 04 2023

Extensions

A-number typo for sqrt(Pi) corrected by R. J. Mathar, Aug 01 2010

A273086 Decimal expansion of theta_3(0, exp(-sqrt(6)*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Offset: 1

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Author

Vaclav Kotesovec, May 14 2016

Keywords

nonn
cons

Examples

1.0009099218872567629192860041215666718045881467303013308592...

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Eric Weisstein's MathWorld, Jacobi Theta Functions

Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A086231, A273087.

Programs

Maple

evalf(((6 + sqrt(6*(3 + 2*sqrt(2)))) * GAMMA(1/24) * GAMMA(5/24) * GAMMA(7/24) * GAMMA(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)), 120); evalf((4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(GAMMA(1/24)*GAMMA(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[6]*Pi]], 10, 105][[1]] RealDigits[((6 + Sqrt[6*(3 + 2*Sqrt[2])]) * Gamma[1/24] * Gamma[5/24] * Gamma[7/24] * Gamma[11/24])^(1/4) / (2*6^(3/8)*Pi^(3/4)), 10, 105][[1]] RealDigits[(4 - Sqrt[2] + Sqrt[6])^(1/4) * Sqrt[Gamma[1/24]*Gamma[11/24]] / (2^(3/2)*3^(3/8)*Pi^(3/4)), 10, 105][[1]]

PARI

th3(x)=1 + 2*suminf(n=1,x^n^2) th3(exp(-sqrt(6)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals ((6 + sqrt(6*(3 + 2*sqrt(2)))) * Gamma(1/24) * Gamma(5/24) * Gamma(7/24) * Gamma(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)).

Equals (4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(Gamma(1/24)*Gamma(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)).

A248557 Decimal expansion of (Pi/2)^(1/4)/Gamma(3/4).

Original entry on oeis.org

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

Offset: 0

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Author

Jean-François Alcover, Dec 15 2014

Keywords

nonn
cons
easy

Examples

0.913579138156116821407242593401222089701963916393469...

Links

MathOverflow, How to calculate the infinite sum of this double series?, 2014.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Crossrefs

Cf. A010767, A068465, A092040, A175573, A222068, A251992.

Programs

Mathematica

RealDigits[(Pi/2)^(1/4)/Gamma[3/4], 10, 103] // First

PARI

(Pi/2)^(1/4)/gamma(3/4) \\ Michel Marcus, Dec 15 2014

Formula

Also equals theta_2(0,exp(-Pi)), where 'theta' is the elliptic theta function.

Equals A175573 / exp(4*A251992/Pi + Pi/4).

Equals Product_{k>=1} tanh(k*Pi/2). - Amiram Eldar, Jun 12 2021

A273087 Decimal expansion of theta_3(0, exp(-sqrt(2)*Pi)), where theta_3 is the 3rd Jacobi theta function.

Original entry on oeis.org

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

Offset: 1

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Author

Vaclav Kotesovec, May 14 2016

Keywords

nonn
cons

Examples

1.0235239993410058634977986567249718525649146079487847418721...

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Eric Weisstein's MathWorld, Jacobi Theta Functions

Wikipedia, Theta function

Crossrefs

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A273086.

Programs

Magma

C := ComplexField(); Gamma(1/8)/(2^(9/8)*Sqrt(Pi(C)*Gamma(1/4))) // G. C. Greubel, Jan 07 2018

Maple

evalf(GAMMA(1/8)/(2^(9/8)*sqrt(Pi*GAMMA(1/4))), 120);

Mathematica

RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[2]*Pi]], 10, 105][[1]] RealDigits[Gamma[1/8]/(2^(9/8)*Sqrt[Pi*Gamma[1/4]]), 10, 105][[1]]

PARI

th3(x)=1 + 2*suminf(n=1,x^n^2) th3(exp(-sqrt(2)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

Formula

Equals Gamma(1/8)/(2^(9/8)*sqrt(Pi*Gamma(1/4))).

A317651 Sequence related to the Taylor expansion of the Jacobi theta_3 constant.

Original entry on oeis.org

1, 1, -1, 51, 849, -26199, 1341999, 82018251, 18703396449, -993278479599, -78795859032801, 38711746282537251, -923351332174412751, 4688204953344642495801, 501271295036889289819599, -89944302490128540556106949, -104694993963067299023875442751, 63396004159664562363095882996001

Offset: 0

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Author

Michel Marcus, Aug 03 2018

Keywords

sign

Links

Christian Krattenthaler and Thomas W. Müller, The congruence properties of Romik's sequence of Taylor.coefficients of Jacobi's theta function theta_3, arXiv:2304.11471 [math.NT], 2023. See p. 1.

Dan Romik, The Taylor coefficients of the Jacobi theta_3, arXiv:1807.06130 [math.NT], 2018.

Robert Scherer, Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant theta_3, arXiv:1904.04509 [math.NT], 2019.

Tanay Wakhare, Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019.

Crossrefs

Cf. A175573 (theta3(1)).

Extensions

More terms from Romik article added by Michel Marcus, Apr 10 2019

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A247217 Decimal expansion of theta_3(0, exp(-2*Pi)), where theta_3 is the 3rd Jacobi theta function.

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A273081 Decimal expansion of theta_3(0, exp(-3*Pi)), where theta_3 is the 3rd Jacobi theta function.

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A273082 Decimal expansion of theta_3(0, exp(-4*Pi)), where theta_3 is the 3rd Jacobi theta function.

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Magma

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A273083 Decimal expansion of theta_3(0, exp(-5*Pi)), where theta_3 is the 3rd Jacobi theta function.

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A273084 Decimal expansion of theta_3(0, exp(-6*Pi)), where theta_3 is the 3rd Jacobi theta function.

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Magma

Maple

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PARI

Formula

A175574 Decimal expansion of sqrt(Pi) / (Gamma(3/4))^2.

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MATLAB

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Extensions