A143233 - OEIS

%I #32 Dec 15 2024 07:20:58

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

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

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

%N Decimal expansion of the dimer constant.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.23, p. 407.

%H G. C. Greubel, <a href="/A143233/b143233.txt">Table of n, a(n) for n = 0..10000</a>

%H Jesús Guillera and Jonathan Sondow, <a href="https://arxiv.org/abs/math/0506319">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, arXiv:math/0506319 [math.NT], 2005-2006; Ramanujan J., Vol. 16 (2008), pp. 247-270; see Example 5.5.

%H Yong Kong, <a href="http://arxiv.org/abs/cond-mat/0610690">Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density</a>, arXiv:cond-mat/0610690, 2006.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominoTiling.html">Domino Tiling</a>

%F Equals Catalan/Pi.

%F Also equals integral_{t=-Pi..Pi} arccosh(sqrt(cos(t)+3)/sqrt(2))/(4*Pi) dt. - _Jean-François Alcover_, May 14 2014

%F From _Antonio Graciá Llorente_, Oct 11 2024: (Start)

%F Equals Sum_{n>=0} (n/2^(n + 2)) * Sum_{k>=0} (-1)^(k + 1)*binomial(n, k)*log(2*k + 1), (Guillera and Sondow, 2008).

%F Equals Sum_{n>=1} n*(arccoth((4*n)/3) - 3*arccoth(4*n)). (End)

%F Equals A006752/Pi = log(A097469) = 2*A322757. - _Hugo Pfoertner_, Oct 11 2024

%e 0.29156090403081878013...

%t RealDigits[Catalan/Pi, 10, 100][[1]] (* _G. C. Greubel_, Aug 24 2018 *)

%o (PARI) default(realprecision, 100); Catalan/Pi \\ _G. C. Greubel_, Aug 24 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Catalan(R)/Pi(R); // _G. C. Greubel_, Aug 24 2018

%Y Cf. A000796, A006752, A097469, A322757.

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Jul 31 2008