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 A127866 #13 Aug 19 2024 08:37:44
%S A127866 4,12,52,172,580,1852,5828,17980,54788,165116,493316,1463036,4312068,
%T A127866 12641276,36887556,107201532,310427652,896045052,2579017732,
%U A127866 7403843580,21205303300,60604891132,172872744964,492233179132,1399272374276
%N A127866 Number of L-shaped 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 A127866 P. Chinn, R. Grimaldi and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Heubach/heubach40.html">Tiling with L's and Squares</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
%H A127866 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3, 4, -8, -12, -4).
%F A127866 a(n) = 4 (-1)^n - (2/9)[(9-5*Sqrt(3))(1+Sqrt(3))^n + (9+5*Sqrt(3))(1-Sqrt(3))^n] - (n/3)[(1-Sqrt(3))(1+Sqrt(3))^n+ (1+Sqrt(3))(1-Sqrt(3))^n].
%F A127866 G.f.: 4x^2/((1+x)(1-2x-2x^2)^2).
%e A127866 a(2) = 4 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 4 L-shaped tiles.
%t A127866 Table[Coefficient[Normal[Series[4x^2/((1 + x)(1 - 2x - 2x^2)^2), {x, 0, 20}]], x, n], {n, 0, 20}]
%Y A127866 Cf. A127864, A127865, A127867, A127868, A127869, A127870, A127871, A127872.
%K A127866 easy,nonn
%O A127866 2,1
%A A127866 Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
%E A127866 G.f. proposed by Maksym Voznyy checked and corrected by _R. J. Mathar_, Sep 16 2009.