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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).
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%I #70 Nov 03 2023 16:18:14

%S 1,1,3,29,901,89893,28793575,29607089625,97725875584681,

%T 1035449388414303593,35216739783694029601963,

%U 3844747107219467355553841461,1347358497824862447450096142795629,1515633798331963142551890627742773295309

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

%H 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 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 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 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 Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, Tianyuan Xu, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p66">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 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 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 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 G.f. = 1 + x + 3*x^2 + 29*x^3 + 901*x^4 + 89893*x^5 + 28793575*x^6 + ...

%t 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 Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Nov 11 2018 *)

%t 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 Cf. A099390, A004003, A256043.

%K nonn

%O 0,3

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

%E a(0)=1 prepended by _Alois P. Heinz_, Mar 25 2015