A118292 -id:A118292 - cp's OEIS frontend

A073005 Decimal expansion of Gamma(1/3).

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and exp(sqrt(3)*Pi) over Q. - Charles R Greathouse IV, Nov 11 2013

			Gamma(1/3) = 2.6789385347077476336556929409746776441286893779573011009...

H. B. Dwight, Tables of Integrals and other Mathematical Data. 860.18, 860.19 in Definite Integrals. New York, U.S.A.: Macmillan Publishing, 1961, p. 230.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 8.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
Andrea Pinos, Gamma of reciprocal by Laplace.
Simon Plouffe, GAMMA(1/3).
Wikipedia, Particular values of the Gamma function: General rational arguments.
Index entries for transcendental numbers.
Index to sequences related to the Gamma function.

Cf. A073006, A202623, A203145, A256165.

R:= RealField(100); SetDefaultRealField(R); Gamma(1/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[1/3], 110]][[1]]
```

default(realprecision, 1080); x=gamma(1/3); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b073005.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009

this * A073006 = A186706. - R. J. Mathar, Jan 15 2021
From Amiram Eldar, Jun 25 2021: (Start)
Equals 2^(7/9) * Pi^(1/3) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/3)/3^(1/12), where K is the complete elliptic integral of the first kind.
Equals 2^(7/9) * Pi^(2/3) /(AGM(2, sqrt(2+sqrt(3)))^(1/3) * 3^(1/12)), where AGM is the arithmetic-geometric mean. (End)
From Andrea Pinos, Aug 12 2023: (Start)
Equals Integral_{x=0..oo} 3*exp(-(x^3)) dx = 3*A202623.
General result: Gamma(1/n) = Integral_{x=0..oo} n*exp(-(x^n)) dx. (End)
Equals 3*A202623 = exp(A256165). - Hugo Pfoertner, Jun 28 2024
Equals (2^(1/3)*Pi*C*3^(1/2))^(1/3), where C = A118292 = Integral {0..1} 2/sqrt(1-x^3) is the transcendental butterfly constant. - Jan Lügering, Feb 08 2025

A146752 a(n) = numerator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).

Original entry on oeis.org

1, 7, 71, 1159, 5197, 148025, 730141, 29616293, 125438657, 1319937329, 77390680651, 76972298827, 319946679037, 3504590799071, 289784158718029, 25703039917515461, 1114069690728835, 112203290640603311

Offset: 0

Artur Jasinski, Nov 01 2008

Previous name was: a(n) is the numerator of k_n such that Integral_{x=0..1} ((1+x^(3n))/sqrt(1-x^3)) dx = k_n*Gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi) for n >= 0.

General formula: Integral_{x=0..1} ((1+x^(3n))/sqrt(1-x^3)) dx = G_3 * k_n = G_3*A146752(n)/A146753(n) = A118292*A146752(n)/A146753(n) where G_3 = (Gamma(1/3)^3)/(2^(1/3)*sqrt(3)*Pi).

Cf. A146753 (denominator), A118292 (G_3).

Table[Numerator[(1/2) (1 + Product[(2 (1 + 3 k))/(5 + 6 k), {k, 0, n - 1}])], {n, 0, 30}]

a(n) = numerator((1/2)*(1 + Product_{k=0..n-1} 2*(1 + 3*k)/(5 + 6*k))).

Simpler name (using given formula) from Joerg Arndt, Sep 24 2022

A146753 a(n) = denominator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).

Original entry on oeis.org

1, 10, 110, 1870, 8602, 249458, 1247290, 51138890, 218502530, 2316126818, 136651482262, 136651482262, 570720896506, 6277929861566, 521068178509978, 46375067887388042, 2016307299451654, 203647037244617054

Offset: 0

Artur Jasinski, Nov 01 2008

Previous name was: a(n)=denominator of k_n such that Integrate[(1+x^(3n))/Sqrt[1-x^3],{x,0,1}]= k_n*(Gamma[1/3]^3)/(2^(1/3)Sqrt[3]Pi) where n >= 0.

General formula: Integral_{x=0..1} ((1+x^(3n))/sqrt(1-x^3)) dx = G_3 * k_n = G_3*A146752(n)/A146753(n) = A118292*A146752(n)/A146753(n) where G_3 = (Gamma(1/3)^3)/(2^(1/3)*sqrt(3)*Pi).

Cf. A146752 (numerator), A118292 (G_3).

Table[Denominator[(1/2) (1 + Product[(2 (1 + 3 k))/(5 + 6 k), {k, 0, n - 1}])], {n, 0, 30}]

a(n) = denominator((1/2)*(1 + Product_{k=0..n-1} 2*(1 + 3*k)/(5 + 6*k))).

New name (using given formula) from Joerg Arndt, Sep 24 2022

A160323 Continued fraction for (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).

Original entry on oeis.org

2, 1, 4, 8, 1, 27, 1, 19, 1, 25, 3, 6, 4, 1, 37, 1, 1, 7, 1, 75, 1, 13, 1, 2, 6, 1, 16, 1, 6, 2, 1, 1, 3, 1, 5, 3, 36, 1, 4, 17, 1, 2, 1, 1, 1, 12, 1, 1, 7, 1, 3, 1, 10, 13, 3, 7, 3, 1, 9, 206, 1, 1, 1, 3, 34, 1, 10, 1, 1, 7, 1, 705, 1, 4, 4, 1, 1, 2, 1, 4, 2, 2, 1, 3, 8, 1, 19, 2, 1, 11, 3, 1, 725, 1, 37

Offset: 0

Harry J. Smith, May 09 2009

gamma(1/6)*gamma(1/3)/(3*sqrt(Pi)) = gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi).

			2.804364210650908522350038158... = 2 + 1/(1 + 1/(4 + 1/(8 + 1/(1 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..3999

Cf. A118292 (decimal expansion).

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction((Gamma(1/6)*Gamma(1/3))/(3*Sqrt(Pi(R)))); // G. C. Greubel, Oct 05 2018

ContinuedFraction[(Gamma[1/6]*Gamma[1/3])/(3*Sqrt[Pi]), 100] (* G. C. Greubel, Oct 05 2018 *)

{ allocatemem(932245000); default(realprecision, 4100); x=gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi); x=contfrac(x); for (n=1, 4000, write("b160323.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Jun 20 2009

Offset changed by Andrew Howroyd, Aug 09 2024

A118811 Decimal expansion of arc length of the (first) butterfly curve.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 30 2006

			9.0173569856546976918...

Eric Weisstein's World of Mathematics, Butterfly Curve

Cf. A118292.

eq = y^6 == x^2-x^6; f[x_] = y /. Solve[eq, y][[2]]; g[y_] = x /. Solve[eq, x][[2]]; h[y_] = x /. Solve[eq, x][[4]]; x1 = 3/8; y1 = f[x1]; x2 = 7/8; y2 = f[x2]; ni[a_, b_] := NIntegrate[a, b, WorkingPrecision -> 120]; i1 = ni[Sqrt[1+f'[x]^2], {x, x1, x2}]; i2 = ni[Sqrt[1+g'[y]^2], {y, 0, y2}]; i3 = ni[Sqrt[1+h'[y]^2], {y, 0, y1}]; Take[RealDigits[4(i1+i2+i3)][[1]], 99](* Jean-François Alcover, Jan 19 2012 *)

4*intnum(x=0,1,sqrt(1+(x/3-x^5)^2/(x^2-x^6)^(5/3))) \\ Charles R Greathouse IV, Jan 17 2012

Last digit corrected by Eric W. Weisstein, Jan 18 2012

A371859 Decimal expansion of Integral_{x=0..oo} 1 / sqrt(1 + x^5) dx.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.54969627774735302956219538317088212891969758...

Decimal expansions of Integral_{x=0..oo} 1 / sqrt(1 + x^k) dx: A118292 (k=3), A093341 (k=4), this sequence (k=5).

Cf. A001622, A087197, A340723, A352324.

RealDigits[Gamma[3/10] Gamma[6/5]/Sqrt[Pi], 10, 105][[1]]
RealDigits[2^(2/5) * Gamma[1/5]^2 / (5*GoldenRatio*Gamma[2/5]), 10, 105][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

Equals Gamma(3/10) * Gamma(6/5) / sqrt(Pi).

Equals 2^(2/5) * Gamma(1/5)^2 / (5 * phi * Gamma(2/5)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Apr 09 2024

A371860 Decimal expansion of Integral_{x=0..1} 1 / sqrt(1 - x^3) dx.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 09 2024

			1.4021821053254542611750190790502941354630222...

Decimal expansions of Integral_{x=0..1} 1 / sqrt(1 - x^k) dx: A019669 (k=2), this sequence (k=3), A085565 (k=4).

Cf. A002161, A118292, A202623, A203145.

RealDigits[Sqrt[Pi] Gamma[4/3]/Gamma[5/6], 10, 104][[1]]
RealDigits[Gamma[1/3]^3 / (2^(4/3)*Sqrt[3]*Pi), 10, 104][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

Equals sqrt(Pi) * Gamma(4/3) / Gamma(5/6).
Equals Gamma(1/3)^3 / (2^(4/3) * sqrt(3) * Pi). - Vaclav Kotesovec, Apr 09 2024
Equals A118292/2. - Hugo Pfoertner, Apr 09 2024

A371861 Decimal expansion of Integral_{x=0..1} sqrt(1 - x^3) dx.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 09 2024

			0.8413092631952725567050114474301764812778...

Decimal expansions of Integral_{x=0..1} sqrt(1 - x^k) dx: A003881 (k=2), this sequence (k=3), A225119 (k=4).

Cf. A002161, A073005, A118292, A203131.

RealDigits[Sqrt[Pi] Gamma[1/3]/(6 Gamma[11/6]), 10, 103][[1]]
RealDigits[Sqrt[3] * Gamma[1/3]^3 / (5*Pi*2^(4/3)), 10, 103][[1]] (* Vaclav Kotesovec, Apr 09 2024 *)

intnum(x=0, 1, sqrt(1 - x^3)) \\ Michel Marcus, Apr 10 2024

Equals sqrt(Pi) * Gamma(1/3) / (6 * Gamma(11/6)).
Equals sqrt(3) * Gamma(1/3)^3 / (5*Pi*2^(4/3)). - Vaclav Kotesovec, Apr 09 2024
Equals 3*A118292/10. - Hugo Pfoertner, Apr 09 2024

cp's OEIS Frontend

A073005 Decimal expansion of Gamma(1/3).

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A146752 a(n) = numerator((1/2)*(1 + Product_{k=0..n-1} 2*(1 + 3*k)/(5 + 6*k))).

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A146753 a(n) = denominator((1/2)*(1 + Product_{k=0..n-1} 2*(1 + 3*k)/(5 + 6*k))).

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Mathematica

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A160323 Continued fraction for (Gamma(1/6)*Gamma(1/3))/(3*sqrt(Pi)).

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Magma

Mathematica

PARI

Extensions

A118811 Decimal expansion of arc length of the (first) butterfly curve.

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Mathematica

PARI

Extensions

A371859 Decimal expansion of Integral_{x=0..oo} 1 / sqrt(1 + x^5) dx.

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Mathematica

Formula

A371860 Decimal expansion of Integral_{x=0..1} 1 / sqrt(1 - x^3) dx.

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Author

Keywords

Examples

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Programs

Mathematica

Formula

A371861 Decimal expansion of Integral_{x=0..1} sqrt(1 - x^3) dx.

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Author

A146752 a(n) = numerator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).

A146753 a(n) = denominator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).

A160323 Continued fraction for (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).