A073005 -id:A073005 - cp's OEIS frontend

A269559 Decimal expansion of Psi(log(2)), negated.

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

Offset: 1

Iaroslav V. Blagouchine, Feb 29 2016

Psi(x) is the digamma function (logarithmic derivative of the Gamma function).

			-1.2395972796176185082441275516860842454332895226874208...

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

Cf. A269557, A269558, A269545, A269546, A269547, A257955, A257957, A257958, A257959, A002161.

Cf. A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

MATLAB
```
format long; psi(log(2))
```
Maple
```
evalf(Psi(ln(2)), 120);
```

RealDigits[PolyGamma[Log[2]], 10, 120][[1]]

default(realprecision, 120); psi(log(2))

A284868 Decimal expansion of the derivative Ai'(0) (negated), where Ai is the Airy function of the first kind.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 04 2017

			-0.2588194037928067984051835601892039634790911383549345822100018138561...

Simon Plouffe, Airy Functions Constant C2
Eric Weisstein's MathWorld, Airy Functions
Wikipedia, Airy Function

Cf. A096714, A096715, A269892, A269893, A073005 (Gamma(1/3)), A284867 (Ai(0)).

Mathematica
```
RealDigits[AiryAi'[0], 10, 105][[1]]
```

-derivnum(x=0,airy(x)[1]) \\ Charles R Greathouse IV, Apr 26 2019

Ai'(0) = -1/(3^(1/3)*Gamma(1/3)).

A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.7733429420779898501961016...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A010469, A019670, A073005, A073006, A204068, A217481.

evalf(int(cos(x^3),x=0..infinity),120); # Muniru A Asiru, Sep 26 2018

RealDigits[Gamma[1/3]/(2*Sqrt[3]), 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Pi/(3*gamma(2/3)) \\ Gheorghe Coserea, Sep 26 2018

intnum(x=[0, -2/3], [oo, I], cos(x)/x^(2/3))/3 \\ Gheorghe Coserea, Sep 26 2018

Equals Pi/(3*Gamma(2/3)) = A019670 / A073006.

Equals Gamma(1/3)/(2*sqrt(3)) = A073005 / A010469. - Amiram Eldar, May 26 2023

A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 07 2017

			1.766638750285449957313689499648438702571868538202557530126905241835453...

Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.

StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.

Cf. A073005 (Gamma(1/3)), A073006 (Gamma(2/3)), A197374 (Beta(1/3,1/3)).

Cf. A104133, A104134, A197374.

RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]

(1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017

Equals (1/3)*Beta(1/3,1/3).
Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).
Equals A197374/3. - Michel Marcus, Jun 08 2020
From Peter Bala, Mar 01 2022: (Start)
Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.
Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).
Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)

A030651 Continued fraction for Gamma(1/3).

Original entry on oeis.org

2, 1, 2, 8, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 9, 1, 4, 1, 3, 1, 7, 1, 1, 3, 6, 2, 2, 1, 1, 32, 3, 3, 3, 1, 24, 2, 2, 25, 1, 2, 1, 6, 2, 1, 1, 3, 1, 9, 3, 2, 1, 6, 7, 2, 8, 2, 5, 1, 5, 1, 2, 2, 2, 2, 4, 3, 1, 5, 1, 15, 1, 1, 2, 4, 3, 3, 1, 5, 1, 4, 1, 8, 1, 3, 1, 1, 8, 2, 1, 2, 1, 514, 1, 2, 1, 1, 1

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it)

			Gamma(1/3) = 2.67893853470774... = 2 + 1/(1 + 1/(2 + 1/(8 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 20 2009

Harry J. Smith, Table of n, a(n) for n = 0..999
Gang Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A030652, A073005 (decimal expansion).

ContinuedFraction[Gamma[1/3], 50] (* Alonso del Arte, Mar 30 2020 *)

{ default(realprecision, 1080); x=contfrac(gamma(1/3)); for (n=1, 1000, write("b030651.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

Note that 3 * Gamma(1/3) * Gamma(2/3) = 2 * Pi * sqrt(3).

Offset changed by Andrew Howroyd, Aug 03 2024

A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

Imaginary part associated with A204067.

			0.446489755784624605609282...

R. J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (3.8).
Wikipedia, Fresnel Integral.

Cf. A073005, A073010, A073006, A202623, A204067

Maple
```
evalf(Pi/GAMMA(2/3)/3^(3/2) ) ;
```

RealDigits[Gamma[1/3]/6, 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

Equals Pi/(Gamma(2/3)* 3^(3/2)) = A073010 / A073006.
(this value)^2 + A204067^2 = A202623^2.
Equals Gamma(1/3)/6 = A073005 / 6. - Amiram Eldar, May 26 2023

A376859 Decimal expansion of Product_{k=1..4} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 09 2024

			3.23937134071697320618006601163079489801213782...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A019692, A202623.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376911 (m = 5 and m = 6), A376912 (m = 7), A376913 (m = 8).

First[RealDigits[2*Pi*Gamma[4/3]/Sqrt[3], 10, 100]]

Equals 2*Pi*Gamma(1/3)/(3*sqrt(3)) = 2*Pi*Gamma(4/3)/sqrt(3) = A186706*A202623 (cf. eq. 86 in Weisstein link).

A376911 Decimal expansion of Product_{k=1..5} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 11 2024

			2.9243272299524025537287380740373781141670220...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002388, A214549.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376859 (m = 4), A376912 (m = 7), A376913 (m = 8).

Mathematica
```
First[RealDigits[8/27*Pi^2, 10, 100]]
```

Equals Product_{k=1..6} Gamma(k/3) = (8/27)*Pi^2 = (8/27)*A002388 (cf. eqs. 87 and 88 in Weisstein link).

Equals 2*A214549. - Hugo Pfoertner, Oct 11 2024

A376912 Decimal expansion of Product_{k=1..7} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 11 2024

			3.4818190686287359395989520629227422880073368098...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002388.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376859 (m = 4), A376911 (m = 5 and m = 6), A376913 (m = 8).

First[RealDigits[32/243*Pi^2*Gamma[1/3], 10, 100]]

Equals (32/243)*Pi^2*Gamma(1/3) = (32/243)*A002388*A073005 (cf. eq. 89 in Weisstein link).

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 11 2024

			5.2386596251856584103292320999763662681359773992...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A091925.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376859 (m = 4), A376911 (m = 5 and m = 6), A376912 (m = 7).

First[RealDigits[640*Pi^3/(2187*Sqrt[3]), 10, 100]]

Equals 640*Pi^3/(2187*sqrt(3)) = 640*A091925/(3^7*A002194) (cf. eq. 90 in Weisstein link).

cp's OEIS Frontend

A269559 Decimal expansion of Psi(log(2)), negated.

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A284868 Decimal expansion of the derivative Ai'(0) (negated), where Ai is the Airy function of the first kind.

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A204067 Decimal expansion of the Fresnel Integral, Integral_{x >= 0} cos(x^3) dx.

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A290570 Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).

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A030651 Continued fraction for Gamma(1/3).

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A204068 Decimal expansion of the Fresnel Integral Integral_{x>=0} sin(x^3) dx.

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A376859 Decimal expansion of Product_{k=1..4} Gamma(k/3).

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