A218387 Decimal expansion of the spanning tree constant of the square lattice.

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

Offset: 1

Author

R. J. Mathar, Oct 27 2012

Keywords

• cons
• easy
• nonn

Examples

			1.16624361612327512055353782587357967545626461594...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 5.22.6, pp. 54, 399.
• Asmus L. Schmidt, Ergodic theory of complex continued fractions, Number Theory with an Emphasis on the Markoff Spectrum, in: A. D. Pollington and W. Moran (eds.), Number Theory with an Emphasis on the Markoff Spectrum, Dekker, 1993, pp. 215-226.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Anthony J. Guttmann, Spanning tree generating functions and Mahler measure, arXiv:1207.2815 [math-ph], 2012.
• Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556, L*(1).

Crossrefs

Cf. A006752 (Catalan), A088538 (4/Pi), A229728, A247685.

Programs

Magma

R:= RealField(100); 4*Catalan(R)/Pi(R); // G. C. Greubel, Aug 23 2018

Maple

evalf(Catalan*4/Pi) ;

Mathematica

RealDigits[4*Catalan/Pi, 10, 100][[1]] (* G. C. Greubel, Aug 23 2018 *)

PARI

default(realprecision, 100); 4*Catalan/Pi \\ G. C. Greubel, Aug 23 2018

Formula

Equals the product of A006752 by A088538.

From Amiram Eldar, Jul 22 2020: (Start)

Equals 1 + Sum_{k>=1} (2*k-1)!!^2/((2*k)!!^2 * (2*k + 1)).

Equals Sum_{k>=0} binomial(2*k,k)^2/(16^k * (2*k + 1)). (End)

Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)) [Schmidt] (see Finch). - Stefano Spezia, Nov 07 2024

Equals log(A229728) = A247685/Pi. - Hugo Pfoertner, Nov 07 2024

Equals Integral_{x=0..1} EllipticK(x)/(Pi*sqrt(x)) dx. - Kritsada Moomuang, Jun 21 2025