A073005 -id:A073005 - cp's OEIS frontend

A355178 Decimal expansion of 2^(-2/3)/L, where L is the conjectured Landau's constant A081760.

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

Offset: 1

Stefano Spezia, Sep 23 2022

			1.159595266963928365769992051570020881945...

Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See p. 23.

Cf. A002580, A073005, A081760, A175379, A203145, A235362, A357318.

First[RealDigits[N[2^(1/3)*Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]), 90]]]

Equals A002580*A357318.
Equals A235362/A081760 = A235362*A175379/(A073005*A203145).
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2)).
From Gerry Martens, Jul 29 2023: (Start)
Equals hypergeom([1/3, 2/3], [1], 1/2).
Equals sqrt(Pi)/(Gamma(2/3)*Gamma(5/6)). (End)

A357319 Decimal expansion of 6PiGamma(2/3)^2/(sqrt(3)*Gamma(1/3)^4).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Sep 23 2022

			0.3874382387848854205695648847540189480496...

Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 21.

Cf. A000796, A002194, A073005, A073006, A212003.

(8*Pi^3)/(sqrt(3)*GAMMA(1/3)^6): evalf(%, 92); # Peter Luschny, Sep 24 2022

First[RealDigits[N[6*Pi*Gamma[2/3]^2/(Sqrt[3]*Gamma[1/3]^4), 90]]]

6*Pi*gamma(2/3)^2/(sqrt(3)*gamma(1/3)^4) \\ Michel Marcus, Sep 24 2022

Equals 6*A000796*A073006^2/(A002194*A073005^4).

Equals (8*Pi^3)/(sqrt(3)*Gamma(1/3)^6) = A212003/(A002194*A073005^6). - Peter Luschny, Sep 24 2022

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

A372047 Decimal expansion of sqrt(3)*Gamma(1/3)/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 22 2024

			1.160014413116984775294152416942826...

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p. 135, eq. (4.3.7).

Cf. A073005, A120011.

Maple
```
sqrt(3)/4*GAMMA(1/3) ; evalf(%) ;
```

RealDigits[Sqrt[3] * Gamma[1/3]/4, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)

sqrt(3) * gamma(1/3) / 4 \\ Amiram Eldar, Jun 10 2024

Equals Integral_{x=0..oo} sin(x^3)/x^3 dx = A120011 * A073005.

A379092 Decimal expansion of sqrt(A224273).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 15 2024

			1.2087177032733153272448479435148561634349207331...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24.1, p. 413.

R. J. Baxter, Colorings of a hexagonal lattice, Journal of Mathematical Physics, Vol. 11, No. 3 (1970), pp. 784-789.
R. J. Baxter, q colourings of the triangular lattice, Journal of Physics A: Mathematical and General, Vol. 19, No. 14 (1986), pp. 2821-2839.
Eric Weisstein's World of Mathematics, Baxter's Four-Coloring Constant.

Cf. A004117, A019692, A073005, A224273.

RealDigits[Sqrt[3] Gamma[1/3]^(3/2)/(2 Pi), 10, 100][[1]]

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

cp's OEIS Frontend

A355178 Decimal expansion of 2^(-2/3)/L, where L is the conjectured Landau's constant A081760.

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A357319 Decimal expansion of 6*Pi*Gamma(2/3)^2/(sqrt(3)*Gamma(1/3)^4).

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

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A372047 Decimal expansion of sqrt(3)*Gamma(1/3)/4.

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A379092 Decimal expansion of sqrt(A224273).

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A357319 Decimal expansion of 6PiGamma(2/3)^2/(sqrt(3)*Gamma(1/3)^4).