This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143233 #54 Sep 02 2025 04:06:13
%S A143233 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 A143233 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 A143233 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 A143233 Decimal expansion of the dimer constant.
%D A143233 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.23, p. 407.
%H A143233 G. C. Greubel, <a href="/A143233/b143233.txt">Table of n, a(n) for n = 0..10000</a>
%H A143233 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 A143233 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 A143233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominoTiling.html">Domino Tiling</a>.
%F A143233 Equals Catalan/Pi = A006752/A000796.
%F A143233 Equals Integral_{t=-Pi..Pi} arccosh(sqrt(cos(t)+3)/sqrt(2))/(4*Pi) dt. - _Jean-François Alcover_, May 14 2014
%F A143233 From _Antonio Graciá Llorente_, Oct 11 2024: (Start)
%F A143233 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 A143233 Equals Sum_{n>=1} n*(arccoth((4*n)/3) - 3*arccoth(4*n)). (End)
%F A143233 Equals A006752/Pi = log(A097469) = 2*A322757. - _Hugo Pfoertner_, Oct 11 2024
%F A143233 Equals Integral_{x=0..1} EllipticK(x)/(4*Pi*sqrt(x)) dx. - _Kritsada Moomuang_, Jun 04 2025
%F A143233 From _Peter Bala_, Jul 29 2025: (Start)
%F A143233 Equals Sum_{k >= 0} (1/4)^(2*k+1) * binomial(2*k, k)^2/(2*k + 1), a slowly converging series due to Ramanujan. For example, define s(n) = Sum_{k = 0..n} (1/4)^(2*k+1) * binomial(2*k, k)^2/(2*k + 1). Then s(50) = 0.29(07...) is only correct to 2 decimal places.
%F A143233 Define S(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n,k) * binomial(n+k,k) * s(n+k). It appears that S(n) tends to the dimer constant far more rapidly. For example, S(50) = 0.291560904030818780138384456468 39(36...) is correct to 32 decimal places. (End)
%F A143233 Equals Sum_{k>=0} (2^(2*k+1)-4)*zeta(2*k)/(4^(2*k+1)*(2*k+1)). - _Amiram Eldar_, Sep 02 2025
%e A143233 0.29156090403081878013...
%p A143233 evalf[140](Catalan/Pi); # _Alois P. Heinz_, Jun 04 2025
%t A143233 RealDigits[Catalan/Pi, 10, 100][[1]] (* _G. C. Greubel_, Aug 24 2018 *)
%o A143233 (PARI) default(realprecision, 100); Catalan/Pi \\ _G. C. Greubel_, Aug 24 2018
%o A143233 (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Catalan(R)/Pi(R); // _G. C. Greubel_, Aug 24 2018
%Y A143233 Cf. A000796, A006752, A097469, A322757.
%K A143233 nonn,cons,changed
%O A143233 0,1
%A A143233 _Eric W. Weisstein_, Jul 31 2008