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

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.

Also 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

Example

0.29156090403081878013...

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.

Sequence in context: A244807 A021347 A011278 * A266264 A011216 A096250

Adjacent sequences: A143230 A143231 A143232 * A143234 A143235 A143236

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 31 2008

Status

approved