A065072 Number of ways to tile a square of side 2n by dominoes (rectangles of size 2 X 1 or 1 X 2) is 2^n * a(n)^2 (see A004003).

1, 1, 3, 29, 901, 89893, 28793575, 29607089625, 97725875584681, 1035449388414303593, 35216739783694029601963, 3844747107219467355553841461, 1347358497824862447450096142795629, 1515633798331963142551890627742773295309

Offset: 0

Author

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br), Nov 08 2001

Keywords

• nonn

Comments

A099390 is the main entry for this problem. - N. J. A. Sloane, Mar 15 2015

Examples

			G.f. = 1 + x + 3*x^2 + 29*x^3 + 901*x^4 + 89893*x^5 + 28793575*x^6 + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50 (terms n=1..25 from T. D. Noe)
• N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014.
• H. Cohn, 2-adic behavior of numbers of domino tilings, arXiv:math/0008222 [math.CO], 2000.
• Philippe Di Francesco, Twenty Vertex model and domino tilings of the Aztec triangle, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.
• Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015. [Mentions this sequence together with a different sequence (A256043) with the same initial terms]
• Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, Discrete Math. 216 (2000), no. 1-3, 211-219. [_N. J. A. Sloane_, Feb 06 2012]
• James Propp, Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds, Integers (2023) Vol. 23, Art. A30.

Crossrefs

Cf. A099390, A004003, A256043.

Programs

Mathematica

a[n_] := With[{L = 2n}, Sqrt[Product[4 Cos[p Pi/(L+1)]^2 + 4 Cos[q Pi/(L+1)]^2, {p, 1, L/2}, {q, 1, L/2}]/2^(L/2)] // Round];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 11 2018 *)
Table[Resultant[ChebyshevU[2*n, x/2], ChebyshevU[2*n, I*x/2], x]^(1/4) / 2^(n/2), {n, 0, 15}] (* Vaclav Kotesovec, Dec 30 2020 *)

Formula

a(n) ~ exp(G*(2*n + 1)^2/(2*Pi)) / (2^((n-1)/2) * (1 + sqrt(2))^(n + 1/2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 14 2020, updated Dec 30 2020

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 25 2015