A143233 - OEIS

A143233

Decimal expansion of the dimer constant.

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, 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, 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

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OFFSET

0,1

REFERENCES

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000

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

Yong Kong, Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density, arXiv:cond-mat/0610690, 2006.

Eric Weisstein's World of Mathematics, Domino Tiling.

FORMULA

Equals Catalan/Pi = A006752/A000796.

Equals Integral_{t=-Pi..Pi} arccosh(sqrt(cos(t)+3)/sqrt(2))/(4*Pi) dt. - Jean-François Alcover, May 14 2014

From Antonio Graciá Llorente, Oct 11 2024: (Start)

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).

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

Equals A006752/Pi = log(A097469) = 2*A322757. - Hugo Pfoertner, Oct 11 2024

Equals Integral_{x=0..1} EllipticK(x)/(4*Pi*sqrt(x)) dx. - Kritsada Moomuang, Jun 04 2025

From Peter Bala, Jul 29 2025: (Start)

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.

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)

Equals Sum_{k>=0} (2^(2*k+1)-4)*zeta(2*k)/(4^(2*k+1)*(2*k+1)). - Amiram Eldar, Sep 02 2025

EXAMPLE

0.29156090403081878013...

MAPLE

evalf[140](Catalan/Pi); # Alois P. Heinz, Jun 04 2025

MATHEMATICA

RealDigits[Catalan/Pi, 10, 100][[1]] (* G. C. Greubel, Aug 24 2018 *)

PROG

(PARI) default(realprecision, 100); Catalan/Pi \\ G. C. Greubel, Aug 24 2018

(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Catalan(R)/Pi(R); // G. C. Greubel, Aug 24 2018

CROSSREFS

Cf. A000796, A006752, A097469, A322757.