A218387 - OEIS

%I #28 Dec 15 2024 07:20:53

%S 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,

%T 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,

%U 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

%N Decimal expansion of the spanning tree constant of the square lattice.

%D 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.

%D 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.

%H G. C. Greubel, <a href="/A218387/b218387.txt">Table of n, a(n) for n = 1..10000</a>

%H Anthony J. Guttmann, <a href="http://arxiv.org/abs/1207.2815">Spanning tree generating functions and Mahler measure</a>, arXiv:1207.2815 [math-ph], 2012.

%H Sheldon Yang, <a href="https://dx.doi.org/10.1080/0020739X.1192.10715688">Some properties of Catalan's constant G</a>, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556, L*(1).

%F Equals the product of A006752 by A088538.

%F From _Amiram Eldar_, Jul 22 2020: (Start)

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

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

%F 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

%F Equals log(A229728) = A247685/Pi. - _Hugo Pfoertner_, Nov 07 2024

%e 1.16624361612327512055353782587357967545626461594...

%p evalf(Catalan*4/Pi) ;

%t RealDigits[4*Catalan/Pi, 10, 100][[1]] (* _G. C. Greubel_, Aug 23 2018 *)

%o (PARI) default(realprecision, 100); 4*Catalan/Pi \\ _G. C. Greubel_, Aug 23 2018

%o (Magma) R:= RealField(100); 4*Catalan(R)/Pi(R); // _G. C. Greubel_, Aug 23 2018

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

%K cons,easy,nonn

%O 1,3

%A _R. J. Mathar_, Oct 27 2012