%I #45 Feb 16 2025 08:33:19

%S 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,

%T 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,

%U 2,1,3,5,5,8,8,5,7,3,1,4,4,0,7,7,6,5,3

%N Decimal expansion of Baxter's four-coloring constant.

%C The constant is named after Australian physicist Rodney James Baxter. - _Amiram Eldar_, Aug 13 2020

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

%H Vincenzo Librandi, <a href="/A224273/b224273.txt">Table of n, a(n) for n = 1..5000</a>

%H R. J. Baxter, <a href="https://doi.org/10.1063/1.1665210">Colorings of a hexagonal lattice</a>, Journal of Mathematical Physics, Vol. 11, No. 3 (1970), pp. 784-789.

%H R. J. Baxter, <a href="https://doi.org/10.1088/0305-4470/19/14/019">q colourings of the triangular lattice</a>, Journal of Physics A: Mathematical and General, Vol. 19, No. 14 (1986), pp. 2821-2839.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BaxtersFour-ColoringConstant.html">Baxter's Four-Coloring Constant</a>.

%F Equals 1/Product_{n>=1} (1-1/(3n-1)^2) = 3*Gamma(1/3)^3/(4*Pi^2).

%F Equals 1/(2^(1/3)*A081760). - _Kritsada Moomuang_, Mar 15 2020

%F Equals 2*Pi/(sqrt(3)*Gamma(2/3)^3). - _Vaclav Kotesovec_, Mar 23 2020

%F Equals Product_{k>=1} (1 + 1/A152751(k)). - _Amiram Eldar_, Aug 13 2020

%F Equals Sum_{k>=0} binomial(-1/3, k)^2. - _Gerry Martens_, Jul 24 2023

%e 1.46099848620631835815887311784605969703893135580746178820577543...

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

%o (PARI) 3*gamma(1/3)^3/(4*Pi^2) \\ _Michel Marcus_, Mar 23 2020

%Y Cf. A004117, A081760, A152751.

%K nonn,cons,changed

%O 1,2

%A _Bruno Berselli_, Apr 02 2013