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%I #24 Dec 10 2022 07:59:13
%S 1,1,5,11,33,87,241,655,1793,4895,13377,36543,99841,272767,745217,
%T 2035967,5562369,15196671,41518081,113429503,309895169,846649343,
%U 2313089025,6319476735,17265131521,47169216511,128868696065,352075825151,961889042433,2627929735167
%N Number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
%C The signed version of this sequence appears as A077917.
%H Alois P. Heinz, <a href="/A127864/b127864.txt">Table of n, a(n) for n = 0..1000</a>
%H P. Z. Chinn, R. Grimaldi and S. Heubach, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Heubach/heubach40.html">Tiling with Ls and Squares</a>, J. Int. Sequences 10 (2007) #07.2.8.
%H S. Heubach, <a href="https://www.calstatela.edu/sites/default/files/users/u1231/Presentations/talklv.pdf">Tiling with Ls and Squares</a>, 2005.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,2).
%F a(n) = a(n-1) + 4*a(n-2) + 2*a(n-3).
%F a(n) = (-1)^n + (1/sqrt(3)) * ((1+sqrt(3))^n - (1-sqrt(3))^n).
%F G.f.: 1/(1 - x - 4*x^2 - 2*x^3).
%F a(n) = A028860(n+2) + (-1)^n. - _R. J. Mathar_, Oct 29 2010
%F E.g.f.: exp(-x) + (2/sqrt(3))*exp(x)*sinh(sqrt(3)*x). - _G. C. Greubel_, Dec 08 2022
%e a(2) = 5 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile.
%t CoefficientList[Series[1/(1-x-4*x^2-2*x^3), {x,0,30}], x]
%o (Magma) I:=[1,1,5]; [n le 3 select I[n] else Self(n-1) + 4*Self(n-2) + 2*Self(n-3): n in [1..41]]; // _G. C. Greubel_, Dec 08 2022
%o (SageMath)
%o A028860 = BinaryRecurrenceSequence(2,2,-1,1)
%o def A127864(n): return A028860(n+2) + (-1)^n
%o [A127864(n) for n in range(51)] # _G. C. Greubel_, Dec 08 2022
%Y Cf. A077917, A127865, A127866, A127867, A127868, A127869, A127870, A127871, A127872.
%Y Column k=2 of A220054. - _Alois P. Heinz_, Dec 03 2012
%K easy,nonn
%O 0,3
%A Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
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