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

1,3

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).

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

Example

1.16624361612327512055353782587357967545626461594...

Maple

evalf(Catalan*4/Pi) ;

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A191504 A021155 A254245 * A178857 A003676 A369508

Adjacent sequences: A218384 A218385 A218386 * A218388 A218389 A218390

Keyword

cons,easy,nonn

Author

R. J. Mathar, Oct 27 2012

Status

approved