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

1,2

Comments

Finch (2003, p. 232) calls this constant exp(2*Catalan/Pi) "the dimer constant", while A143233 uses the same term for Catalan/Pi.

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.

Eric Weisstein's World of Mathematics, Domino Tiling.

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

Example

1.791622812069593424730547089...

Maple

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

Mathematica

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

Prog

(PARI) exp(2*Catalan/Pi) \\ Charles R Greathouse IV, Jul 15 2014
(Magma) R:=RealField(100); Exp(2*Catalan(R)/Pi(R)); // G. C. Greubel, Aug 23 2018

Crossrefs

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

Cf. A143233 (called the "dimer constant" in OEIS).

Sequence in context: A377622 A086318 A244674 * A132806 A016629 A154203

Adjacent sequences: A130831 A130832 A130833 * A130835 A130836 A130837

Keyword

cons,nonn

Author

R. J. Mathar, Jul 18 2007

Status

approved