cp's OEIS Frontend

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).

%I A065072 #74 Jan 04 2025 07:32:58
%S A065072 1,1,3,29,901,89893,28793575,29607089625,97725875584681,
%T A065072 1035449388414303593,35216739783694029601963,
%U A065072 3844747107219467355553841461,1347358497824862447450096142795629,1515633798331963142551890627742773295309
%N 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).
%C A065072 A099390 is the main entry for this problem. - _N. J. A. Sloane_, Mar 15 2015
%H A065072 Alois P. Heinz, <a href="/A065072/b065072.txt">Table of n, a(n) for n = 0..50</a> (terms n=1..25 from T. D. Noe)
%H A065072 N. Allegra, <a href="http://arxiv.org/abs/1410.4131">Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics</a>, arXiv:1410.4131 [cond-mat.stat-mech], 2014.
%H A065072 H. Cohn, <a href="http://arxiv.org/abs/math/0008222">2-adic behavior of numbers of domino tilings</a>, arXiv:math/0008222 [math.CO], 2000.
%H A065072 Philippe Di Francesco, <a href="https://arxiv.org/abs/2102.02920">Twenty Vertex model and domino tilings of the Aztec triangle</a>, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.
%H A065072 Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, <a href="https://doi.org/10.37236/4472">Sandpiles and Dominos</a>, Electronic Journal of Combinatorics, Volume 22(1), 2015. [Mentions this sequence together with a different sequence (A256043) with the same initial terms]
%H A065072 Peter E. John and Horst Sachs, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00301-5">On a strange observation in the theory of the dimer problem</a>, Discrete Math. 216 (2000), no. 1-3, 211-219. [_N. J. A. Sloane_, Feb 06 2012]
%H A065072 James Propp, <a href="http://math.colgate.edu/~integers/x30/x30.pdf">Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds</a>, Integers (2023) Vol. 23, Art. A30.
%F A065072 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
%e A065072 G.f. = 1 + x + 3*x^2 + 29*x^3 + 901*x^4 + 89893*x^5 + 28793575*x^6 + ...
%t A065072 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];
%t A065072 Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Nov 11 2018 *)
%t A065072 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 *)
%Y A065072 Cf. A099390, A004003, A256043.
%K A065072 nonn
%O A065072 0,3
%A A065072 Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br), Nov 08 2001
%E A065072 a(0)=1 prepended by _Alois P. Heinz_, Mar 25 2015