A038758 -id:A038758 - cp's OEIS frontend

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

Ralf Stephan, Sep 18 2004

			1.33851515197609676693819590201851353706435369712791131464123...

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.

G. C. Greubel, Table of n, a(n) for n = 1..10000
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
J. Propp, Enumeration of matchings: problems and progress, in New Perspectives in Algebraic Combinatorics

Cf. A038758, A054344, A130834, A143233.

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

SetDefaultRealField(RealField(100)); R:=RealField(); Exp(Catalan(R)/Pi(R)); // G. C. Greubel, Aug 25 2018

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

default(realprecision, 100); exp(Catalan/Pi) \\ G. C. Greubel, Aug 25 2018

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

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

A054344 Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes of which exactly 6 are horizontal (or vertical) dominoes.

Original entry on oeis.org

9, 1064, 21656, 197484, 1143366, 4927524, 17240292, 51631617, 137044523, 330284988, 735542444, 1533609350, 3024043008, 5684167992, 10249533240, 17821214019, 30006185613, 49097892704, 78305096016

Offset: 2

Yong Kong (ykong(AT)curagen.com), May 06 2000

			a(3) = 1064 because we have 1064 ways to cover a 36 X 36 lattice with exactly 6 horizontal (or vertical) dominoes and exactly 12 vertical (or horizontal) dominoes.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A004003, A002414, A038758.

CoefficientList[Series[(x^9-10*x^8+45*x^7-36*x^6+3096*x^5 +17256*x^4 +27724*x^3+11421*x^2+974*x+9)/(x-1)^10,{x,0,30}],x] (* Vincenzo Librandi, Jun 26 2012 *)

a(n) = (1/720)*n*(n+1)*(120*n^7-300*n^6-70*n^5+363*n^4+416*n^3-231*n^2-394*n-264).

G.f.: x^2*(x^9-10*x^8+45*x^7-36*x^6+3096*x^5+17256*x^4+27724*x^3+11421*x^2+974*x+9)/(x-1)^10. - Colin Barker, Jun 26 2012

cp's OEIS Frontend

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

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