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.

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

Offset: 1

Author

R. J. Mathar, Jul 18 2007

Keywords

• cons
• nonn

Examples

			1.791622812069593424730547089...
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 232, 407.

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• Steven R. Finch, Several Constants Arising in Statistical Mechanics, Ann. Comb. 3(2-4) (1999), 323-335.
• Antonio Gracia Llorente, Infinite Product Formula Involving the Catalan's Constant, OSF Preprint, 2024.

Crossrefs

Cf. A000796 (Pi), A006752 (Catalan), A097469, A229728.

Programs

Magma

R:=RealField(100); Exp(2*Catalan(R)/Pi(R)); // G. C. Greubel, Aug 23 2018

Maple

evalf(exp(2*Catalan/Pi));

Mathematica

RealDigits[Exp[(2*Catalan)/Pi],10,120][[1]] (* Harvey P. Dale, Jul 17 2011 *)

PARI

exp(2*Catalan/Pi) \\ Charles R Greathouse IV, Jul 15 2014

Formula

Equals exp(2*A006752/A000796).

Equals A097469^2. - Vaclav Kotesovec, Dec 30 2020

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

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