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 A385242 #15 Aug 04 2025 00:58:37
%S A385242 1,0,3,0,12,2,50,16,210,100,888,558,3778,2926,16164,14758,69520,72504,
%T A385242 300458,349586,1304390,1662320,5686114,7821308,24879632,36497742,
%U A385242 109227706,169207550,480982532,780370350,2123682344,3583760736,9398963962,16400994810,41684827750
%N A385242 Number of tilings of a 3 X n strip with dominos and U-shaped pentominos.
%C A385242 Compare to A001835 which counts the tilings of a 3 X 2*(n-1) strip with just dominos. So, there will be 12 tilings of a 3 X 4 strip with dominos and U-shaped pentominos; 11 of them come from the U-free tilings counted in A001835(3), and here is the one additional tiling with two U's:
%C A385242 _______
%C A385242 | _|_ |
%C A385242 | |___| |
%C A385242 |___|___|.
%H A385242 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,0,-2).
%F A385242 G.f.: (x-1)*(x+1)*(x^3+x-1)/(2*x^5+3*x^3-4*x^2-x+1).
%F A385242 a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 2*a(n-5) for n >= 6.
%e A385242 Here are the a(5)=2 ways to tile the 3 X 5 strip with dominos and U's:
%e A385242 _________ _________
%e A385242 |___| |___| | | _ | |
%e A385242 | | |_| | | |_|_| |_|_|
%e A385242 |_|_____|_| |___|_|___|.
%t A385242 Join[{1}, LinearRecurrence[{1, 4, -3, 0, -2}, {0, 3, 0, 12, 2}, 40]]
%Y A385242 Cf. A001835.
%K A385242 nonn,easy
%O A385242 0,3
%A A385242 _Greg Dresden_ and Siting Jia, Jul 28 2025