A081760 -id:A081760 - cp's OEIS frontend

A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.

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

Offset: 0

Stefano Spezia, Sep 23 2022

			0.9203713733179424976555185645431729947262...

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.
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. 2.

Cf. A004117.
Cf. A081760.
Cf. A073005, A175379, A203145.

First[RealDigits[N[Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]),88]]]

1/(2*gamma(1/3)*gamma(5/6)/gamma(1/6)) \\ Michel Marcus, Sep 24 2022

Equals 1/(2*A081760) = A175379/(2*A073005*A203145).
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2))*exp(2*Pi*i*(k/3-m/3)).
Equals Sum_{k>=0} (binomial(-1/3,2*k)^2 - binomial(-1/3,2*k+1)^2). - Gerry Martens, Jul 24 2023
Equals 3*Gamma(1/3)^3 / (2^(8/3) * Pi^2). - Vaclav Kotesovec, Jul 27 2023

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

Original entry on oeis.org

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)

A224273 Decimal expansion of Baxter's four-coloring constant.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Apr 02 2013

The constant is named after Australian physicist Rodney James Baxter. - Amiram Eldar, Aug 13 2020

			1.46099848620631835815887311784605969703893135580746178820577543...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See p. 413.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
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, A081760, A152751.

RealDigits[3 Gamma[1/3]^3/(4 Pi^2), 10, 90][[1]]

3*gamma(1/3)^3/(4*Pi^2) \\ Michel Marcus, Mar 23 2020

Equals 1/Product_{n>=1} (1-1/(3n-1)^2) = 3*Gamma(1/3)^3/(4*Pi^2).
Equals 1/(2^(1/3)*A081760). - Kritsada Moomuang, Mar 15 2020
Equals 2*Pi/(sqrt(3)*Gamma(2/3)^3). - Vaclav Kotesovec, Mar 23 2020
Equals Product_{k>=1} (1 + 1/A152751(k)). - Amiram Eldar, Aug 13 2020
Equals Sum_{k>=0} binomial(-1/3, k)^2. - Gerry Martens, Jul 24 2023

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A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.

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

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A224273 Decimal expansion of Baxter's four-coloring constant.

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