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A130834 Decimal expansion of the limit of the (2/n^2)-th power of the number of distinct dimer coverings on the n X n square grid, n even, as n goes to infinity.

%I A130834 #55 Apr 16 2025 19:39:17
%S A130834 1,7,9,1,6,2,2,8,1,2,0,6,9,5,9,3,4,2,4,7,3,0,5,4,7,0,8,9,3,4,2,9,8,2,
%T A130834 4,3,2,2,6,8,1,3,4,3,9,3,1,3,2,9,5,4,7,6,7,7,6,7,5,8,3,4,7,6,4,9,9,4,
%U A130834 2,5,0,7,4,2,3,7,6,5,7,8,9,6,0,1,3,2,2,6
%N A130834 Decimal expansion of the limit of the (2/n^2)-th power of the number of distinct dimer coverings on the n X n square grid, n even, as n goes to infinity.
%D A130834 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 232, 407.
%H A130834 Nathaniel Johnston, <a href="/A130834/b130834.txt">Table of n, a(n) for n = 1..10000</a>
%H A130834 Steven R. Finch, <a href="http://dx.doi.org/10.1007/BF01608791">Several Constants Arising in Statistical Mechanics</a>, Ann. Comb. 3(2-4) (1999), 323-335.
%H A130834 Antonio Gracia Llorente, <a href="https://osf.io/preprints/osf/rngf6">Infinite Product Formula Involving the Catalan's Constant</a>, OSF Preprint, 2024.
%F A130834 Equals exp(2*A006752/A000796).
%F A130834 Equals A097469^2. - _Vaclav Kotesovec_, Dec 30 2020
%F A130834 Equals Product_{k>=1} (((4*k-1)^3*(4*k+3))/((4*k+1)^3*(4*k-3)))^k. - _Antonio Graciá Llorente_, Jul 22 2024
%F A130834 Equals lim_{n->oo} 1/((4*n)^(2*n))*Product_{k=1..n} ((4*k - 1)^(4*k - 1))/((4*k - 3)^(4*k - 3)). - _Antonio Graciá Llorente_, Apr 16 2025
%e A130834 1.791622812069593424730547089...
%p A130834 evalf(exp(2*Catalan/Pi));
%t A130834 RealDigits[Exp[(2*Catalan)/Pi],10,120][[1]] (* _Harvey P. Dale_, Jul 17 2011 *)
%o A130834 (PARI) exp(2*Catalan/Pi) \\ _Charles R Greathouse IV_, Jul 15 2014
%o A130834 (Magma) R:=RealField(100); Exp(2*Catalan(R)/Pi(R)); // _G. C. Greubel_, Aug 23 2018
%Y A130834 Cf. A000796 (Pi), A006752 (Catalan), A097469, A229728.
%K A130834 cons,nonn
%O A130834 1,2
%A A130834 _R. J. Mathar_, Jul 18 2007