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%I #31 Dec 29 2022 15:11:57
%S 0,1,8,125,1728,24389,343000,4826809,67917312,955671625,13447314152,
%T 189218084021,2662500456000,37464224551181,527161643971768,
%U 7417727240640625,104375343011770368,1468672529408250769
%N a(n) = Pell(n)^3.
%C a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/3,2/3)-fences, black third-squares (1/3 X 1 pieces, always placed so that the shorter sides are horizontal), and white third-squares. A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/6,5/6)-fences, black (1/6,1/3)-fences, and white (1/6,1/3)-fences. - _Michael A. Allen_, Dec 29 2022
%H G. C. Greubel, <a href="/A110272/b110272.txt">Table of n, a(n) for n = 0..850</a>
%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H Toufik Mansour, <a href="https://ajc.maths.uq.edu.au/pdf/30/ajc_v30_p207.pdf">A formula for the generating functions of powers of Horadam's sequence</a>, Australas. J. Combin. 30 (2004) 207-212.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,30,-12,-1).
%F G.f.: x*(1-4*x-x^2) / ((1+2*x-x^2)*(1-14*x-x^2)).
%F a(n) = 12*a(n-1) + 30*a(n-2) - 12*a(n-3) - a(n-4).
%F a(n) = (Pell(3*n) - 3*(-1)^n*Pell(n))/8.
%t Fibonacci[Range[0, 30], 2]^3 (* _G. C. Greubel_, Sep 17 2021 *)
%o (Magma) I:=[0,1,8,125]; [n le 4 select I[n] else 12*Self(n-1) + 30*Self(n-2) -12*Self(n-3) - Self(n-4): n in [1..31]]; // _G. C. Greubel_, Sep 17 2021
%o (Sage) [lucas_number1(n, 2, -1)^3 for n in (0..30)] # _G. C. Greubel_, Sep 17 2021
%Y Cf. A000129, A079291, A213688.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Jul 18 2005
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