A091648 -id:A091648 - cp's OEIS frontend

A257436 Decimal expansion of G(1/3), a generalized Catalan constant.

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

Offset: 0

Jean-François Alcover, Apr 23 2015

			0.855389245838564640972481036740456552226831197315573480398142...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, 2010.
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.

Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257437 (G(1/12)), A257438 (G(1/5)).

SetDefaultRealField(RealField(100)); (3/8)*Sqrt(3)*Log(2 + Sqrt(3)); // G. C. Greubel, Aug 24 2018

RealDigits[(3/8)*Sqrt[3]*Log[2 + Sqrt[3]], 10, 107] // First
N[Pi*HypergeometricPFQ[{1/6, 1/2, 5/6}, {1, 3/2}, 1]/4, 105] (* Vaclav Kotesovec, Apr 24 2015 *)

(3/8)*sqrt(3)*log(2 + sqrt(3)) \\ G. C. Greubel, Aug 24 2018

G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
G(s) = (1/(8*s))*(Pi + cos(Pi*s)*(psi(1/4+s/2) - psi(3/4+s/2))), where psi is the digamma function (PolyGamma).
G(1/3) = (3/8)*sqrt(3)*log(2 + sqrt(3)) = (3/4)*sqrt(3)*arccoth(sqrt(3)).

A257437 Decimal expansion of G(1/12), a generalized Catalan constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 23 2015

			0.91205508937271840001882865569309753445114291198031874684362426...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, 2010.
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.

Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257436 (G(1/3)), A257438 (G(1/5)).

SetDefaultRealField(RealField(100)); 3*(Sqrt(3)+1)*(Log(Sqrt(2)-1) + (Sqrt(3)/2)*Log(Sqrt(3)+ Sqrt(2))) // G. C. Greubel, Aug 24 2018

RealDigits[3*(Sqrt[3]+1)*(Log[Sqrt[2]-1] + (Sqrt[3]/2)*Log[Sqrt[3]+ Sqrt[2]]), 10, 103] // First
N[Pi*HypergeometricPFQ[{5/12, 1/2, 7/12}, {1, 3/2}, 1]/4, 106] (* Vaclav Kotesovec, Apr 24 2015 *)

3*(sqrt(3)+1)*(log(sqrt(2)-1) + (sqrt(3)/2)*log(sqrt(3)+ sqrt(2))) \\ G. C. Greubel, Aug 24 2018

G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
G(s) = (1/(8*s))*(Pi + cos(Pi*s)*(psi(1/4+s/2) - psi(3/4+s/2))), where psi is the digamma function (PolyGamma).
G(1/12) = 3*(sqrt(3)+1)*(log(sqrt(2)-1) + (sqrt(3)/2)*log(sqrt(3)+sqrt(2))).

A257438 Decimal expansion of G(1/5), a generalized Catalan constant.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 23 2015

			0.8936714234609635543020698545835460075475580947963280782203...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, 2010.
D. Borwein, J. M. Borwein, M. L. Glasser, J. G Wan, Moments of Ramanujan's Generalized Elliptic Integrals and Extensions of Catalan's Constant, Journal of Mathematical Analysis and Applications, Volume 384, Issue 2, 15 December 2011, Pages 478-496.

Cf. A006752 (G(0) = Catalan), A257435 (G(1/6)), A091648 (G(1/4)), A257436 (G(1/3)), A257437 (G(1/12)).

SetDefaultRealField(RealField(100)); (5/8)*Sqrt(5+2*Sqrt(5))*(((Sqrt(5)-1)/2)*Argsinh(Sqrt(5+2*Sqrt(5))) - Argsinh(Sqrt(5-2*Sqrt(5)))); // G. C. Greubel, Aug 24 2018

RealDigits[(5/8)*Sqrt[5+2*Sqrt[5]]*(((Sqrt[5]-1)/2)*ArcSinh[Sqrt[5+2*Sqrt[5]]] - ArcSinh[Sqrt[5-2*Sqrt[5]]]), 10, 104] // First
N[Pi*HypergeometricPFQ[{3/10, 1/2, 7/10}, {1, 3/2}, 1]/4, 105] (* Vaclav Kotesovec, Apr 24 2015 *)

(5/8)*sqrt(5+2*sqrt(5))*(((sqrt(5)-1)/2)*asinh(sqrt(5 +2*sqrt(5))) - asinh(sqrt(5-2*sqrt(5)))) \\ G. C. Greubel, Aug 24 2018

G(s) = (Pi/4) * 3F2(1/2, 1/2-s, s+1/2; 1, 3/2; 1), with 2F1 the hypergeometric function.
G(s) = (1/(8*s))*(Pi + cos(Pi*s)*(psi(1/4+s/2) - psi(3/4+s/2))), where psi is the digamma function (PolyGamma).
G(1/5) = (5/8)*sqrt(5+2*sqrt(5))*(((sqrt(5)-1)/2)*arcsinh(sqrt(5+2*sqrt(5))) - arcsinh(sqrt(5-2*sqrt(5)))).

A245592 Decimal expansion of the Ising constant K_c, the ratio of the coupling constant to the ferromagnetic critical temperature, in the two-dimensional case.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 30 2014

			0.4406867935097715126163046624898961545140801641308177053766478...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.22 Lenz-Ising Constants, p. 396.

Andrei Okounkov, Limit shapes, real and imagined, Bull. Amer. Math. Soc. 53 2 (2016), 187-216
Wikipedia, Ising Model.

Cf. A091648.

RealDigits[(1/2)*Log[1 + Sqrt[2]], 10, 102] // First

K_c = (1/2)*log(1 + sqrt(2)) = (1/2)*A091648.
Equals Integral_{0..1} x/sqrt(1 + x^4) dx. - Peter Bala, Oct 28 2019
Equals arcsinh(1)/2. - Peter Luschny, Oct 29 2019
Equals arctanh(sqrt(2)-1). - Amiram Eldar, Feb 09 2024

A093754 Decimal expansion of -Catalan + (Pi*arcsinh(1))/2 + HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

			0.63951035187031100196269308542732367987599462518472...

Eric Weisstein's World of Mathematics, Radial Integrals.

Cf. A000796, A006752, A091648, A093753.

RealDigits[-Catalan + (Pi*ArcSinh[1])/2 + HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6, 10, 100][[1]] (* Amiram Eldar, Apr 12 2022 *)

Equals Integral_{x=0..1, y=0..1} 1/(x^2 + y^2 + 1) dy dx.

Equals Integral_{x=0..1} arccot(sqrt(x^2+1))/sqrt(x^2+1) dx. - Amiram Eldar, Apr 12 2022

Keyword:cons added by R. J. Mathar, Apr 01 2010

A129269 Decimal expansion of arcsinh(1/5).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 27 2008

			0.19869011034924140647463691595020696822130879422445377302126...

Index entries for transcendental numbers

Decimal expansion of arcsinh(1/k): A091648 (k=1), A002390 (k=2), A129187(k=3), A129200 (k=4).

RealDigits[ArcSinh[1/5], 10, 105][[1]] (* Amiram Eldar, May 02 2023 *)

asinh(.2) \\ Charles R Greathouse IV, Nov 21 2024

A157700 Decimal expansion of log(4/(1 + sqrt(2))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 04 2009

Equals Sum_{n>=2, n even} binomial(2n,n)/(n*4^n) = A016627-A091648.

			0.5049207741003475936018...

G. C. Greubel, Table of n, a(n) for n = 0..10000
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly 92, no 7 (1985) 449-457.
Index entries for transcendental numbers

SetDefaultRealField(RealField(100)); Log(4/(1+Sqrt(2))); // G. C. Greubel, Oct 02 2018

Maple
```
log(4/(1+sqrt(2))) ;
```

RealDigits[Log[4/(1+Sqrt[2])],10,120][[1]] (* Harvey P. Dale, Jun 08 2014 *)

default(realprecision, 100); log(4/(1+sqrt(2))) \\ G. C. Greubel, Oct 02 2018

A169800 Decimal expansion of 2/log(1+sqrt(2)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 18 2010

Critical temperature for two-dimensional Ising model.

			2.2691853142130219681144908103065724757259815855040013500506...

G. C. Greubel, Table of n, a(n) for n = 1..10000
M. Richey, The evolution of Markov chain Monte Carlo Methods, Amer. Math. Monthly, 117 (No. 5, 2010), 383-413.
Index entries for transcendental numbers

Cf. A091648 (log(1 + sqrt(2))).

SetDefaultRealField(RealField(100)); 2/Log(1+Sqrt(2)); // G. C. Greubel, Oct 02 2018

RealDigits[2/Log[1 + Sqrt[2]], 10, 100][[1]] (* G. C. Greubel, Oct 02 2018 *)

2/log(1+sqrt(2)) \\ Charles R Greathouse IV, Mar 25 2014

A293812 Decimal expansion of log(3)/log(1 + sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Oct 16 2017

Fractal dimension of the frontier of the Fibonacci word fractal.

			1.24647743572981584189100424874815183996105530003376417796845193354456...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Alexis Monnerot-Dumaine, The Fibonacci Word Fractal, HAL-00367972, February 2009, section 5.2.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]
Index entries for transcendental numbers

Equals A002391 / A091648.

SetDefaultRealField(RealField(105)); n:=Log(1+Sqrt(2),3); Reverse(Intseq(Floor(10^104*n)));

evalf(log(3)/log(1+sqrt(2)),110); # Muniru A Asiru, Oct 11 2018

RealDigits[Log[3]/Log[1 + Sqrt[2]], 10, 100][[1]] (* G. C. Greubel, Oct 10 2018 *)

PARI
```
log(3)/log(1+sqrt(2))
```

A354633 Decimal expansion of the negated digamma function at 3/8.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 01 2022

			psi(3/8) = -2.7539990491451...

Index entries for sequences related to the digamma function

Cf. A001620, A091648.

RealDigits[PolyGamma[3/8], 10, 100][[1]] (* Amiram Eldar, Jun 03 2022 *)

Equals sqrt(2)*arcsinh(1) - 4*log(2) - (sqrt(2)-1)*Pi/2 - gamma, where gamma is Euler's constant (A001620). - Amiram Eldar, Jun 03 2022

cp's OEIS Frontend

A257436 Decimal expansion of G(1/3), a generalized Catalan constant.

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A257437 Decimal expansion of G(1/12), a generalized Catalan constant.

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A257438 Decimal expansion of G(1/5), a generalized Catalan constant.

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A245592 Decimal expansion of the Ising constant K_c, the ratio of the coupling constant to the ferromagnetic critical temperature, in the two-dimensional case.

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A093754 Decimal expansion of -Catalan + (Pi*arcsinh(1))/2 + HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6.

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A129269 Decimal expansion of arcsinh(1/5).

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A157700 Decimal expansion of log(4/(1 + sqrt(2))).

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