This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A127865 #27 Aug 19 2024 08:39:33
%S A127865 2,8,30,108,354,1152,3614,11204,34170,103176,308598,916236,2702834,
%T A127865 7929872,23155182,67333140,195082218,563367960,1622185958,4658753564,
%U A127865 13347741666,38160007200,108881256414,310108078116,881761288154
%N A127865 Number of square tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
%H A127865 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 A127865 S. Heubach, <a href="https://www.calstatela.edu/sites/default/files/users/u1231/Presentations/talklv.pdf">Tiling with Ls and Squares</a>, 2005.
%H A127865 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 7, -4, -20, -16, -4).
%F A127865 a(n) = (2*n - 12)*(-1)^n + (2/3)*((9-5*sqrt(3))*(1+sqrt(3))^n + (9+5*sqrt(3))*(1-sqrt(3))^n) + (n/sqrt(3))*((sqrt(3)-1)*(1+sqrt(3))^n+ (sqrt(3)+1)*(1-sqrt(3))^n).
%F A127865 G.f.: 2*x*(1+2*x)/((1+x)^2*(1-2*x-2*x^2)^2). - _Colin Barker_, Apr 30 2012
%e A127865 a(2) = 8 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 and thus all the tilings of the 2 X 2 board contain 8 square tiles.
%t A127865 Table[(2n - 12)(-1)^n + (2/3)((9 - 5Sqrt[3])(1 + Sqrt[3])^n + (9 + 5Sqrt[3])(1 - Sqrt[3])^n) + (n/Sqrt[3])((Sqrt[3] - 1)( 1 + Sqrt[3])^n + (Sqrt[3] + 1)(1 - Sqrt[3])^n), {n, 1, 30}]
%Y A127865 Cf. A127864, A127866, A127867, A127868, A127869, A127870, A127871, A127872.
%K A127865 easy,nonn
%O A127865 1,1
%A A127865 Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007