A097469 - OEIS

%I #44 Nov 18 2024 04:21:55

%S 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,

%T 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,

%U 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

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

%D 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.

%H G. C. Greubel, <a href="/A097469/b097469.txt">Table of n, a(n) for n = 1..10000</a>

%H F. Ardila and R. P. Stanley, <a href="https://arxiv.org/abs/math/0501170">Tilings</a>, arXiv:math/0501170 [math.CO], 2005.

%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1016/0031-8914(61)90063-5">The Statistics of Dimers on a Lattice</a>, Physica, 27 (1961), 1209-1225.

%H J. Propp, Enumeration of matchings: problems and progress, in <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>

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

%F 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

%F Equals sqrt(A130834) = exp(A143233). - _Hugo Pfoertner_, Nov 18 2024

%e 1.33851515197609676693819590201851353706435369712791131464123...

%t RealDigits[Exp[Catalan/Pi], 10, 100][[1]] (* _G. C. Greubel_, Aug 25 2018 *)

%o (PARI) default(realprecision, 100); exp(Catalan/Pi) \\ _G. C. Greubel_, Aug 25 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Exp(Catalan(R)/Pi(R)); // _G. C. Greubel_, Aug 25 2018

%Y Cf. A038758, A054344, A130834, A143233.

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

%K cons,nonn

%O 1,2

%A _Ralf Stephan_, Sep 18 2004

%E Terms a(14) onward corrected by _G. C. Greubel_, Aug 26 2018