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

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

1,2

Comments

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

References

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

Links

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.

Formula

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

Example

1.46099848620631835815887311784605969703893135580746178820577543...

Mathematica

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

Prog

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

Crossrefs

Cf. A004117, A081760, A152751.

Sequence in context: A200349 A021221 A197006 * A285444 A285584 A316386

Adjacent sequences: A224270 A224271 A224272 * A224274 A224275 A224276

Keyword

nonn,cons

Author

Bruno Berselli, Apr 02 2013

Status

approved