A097469 Decimal expansion of growth constant C for dimer model on square grid.

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

Offset

1,2

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.8.3 and 5.23.1, pp. 63, 407.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Federico Ardila and Richard P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.

Yasuyuki Kachi and Pavlos Tzermias, Infinite Products Involving zeta(3) and Catalan’s Constant, Journal of Integer Sequences, Vol. 15 (2012), Article 12.9.4. See p. 3.

P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.

James Propp, Enumeration of matchings: problems and progress, in New Perspectives in Algebraic Combinatorics, Cambridge University Press, Cambridge, 1999, pp. 255-291.

Formula

Equals e^(G/Pi), with G = A006752 (Catalan's constant).

Equals exp((1/Pi^2) * Integral_{x=0..Pi/2, y=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2) dx dy). - Vaclav Kotesovec, Jan 04 2021

Equals sqrt(A130834) = exp(A143233). - Hugo Pfoertner, Nov 18 2024

Equals Product_{n>=1} (Product_{k=0..n} (2*k+1)^((-1)^k*binomial(n,k)))^(n/(2^(n+2))) (Kachi and Tzermias, 2012). - Amiram Eldar, Jul 14 2026

Example

1.33851515197609676693819590201851353706435369712791131464123...

Mathematica

RealDigits[Exp[Catalan/Pi], 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

Prog

(PARI) default(realprecision, 100); exp(Catalan/Pi) \\ G. C. Greubel, Aug 25 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Exp(Catalan(R)/Pi(R)); // G. C. Greubel, Aug 25 2018

Crossrefs

Cf. A038758, A054344, A130834, A143233.

Cf. A000796 (Pi), A006752 (Catalan's constant).

Sequence in context: A016606 A341616 A069462 * A372409 A279727 A267092

Adjacent sequences: A097466 A097467 A097468 * A097470 A097471 A097472

Keyword

cons,nonn,changed

Author

Ralf Stephan, Sep 18 2004

Extensions

Terms a(14) onward corrected by G. C. Greubel, Aug 26 2018

Status

approved