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 A354080 #46 May 27 2024 15:55:03
%S A354080 1,4,5,10,19,34,63,116,213,392,721,1326,2439,4486,8251,15176,27913,
%T A354080 51340,94429,173682,319451,587562,1080695,1987708,3655965,6724368,
%U A354080 12368041,22748374,41840783,76957198,141546355
%N A354080 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0)=1, a(1)=4, a(2)=5.
%C A354080 a(n) is the number of ways to tile this T-shaped figure of length n with squares, dominoes, and trominoes. Shown here is the figure for n=9.
%C A354080 _
%C A354080 |_|_______________
%C A354080 |_|_|_|_|_|_|_|_|_|
%C A354080 |_|
%H A354080 Paolo Xausa, <a href="/A354080/b354080.txt">Table of n, a(n) for n = 0..1000</a>
%H A354080 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1).
%F A354080 a(n) = T(n+2) + 3*T(n+1), for T(n) = A000073(n) the tribonacci numbers.
%F A354080 a(n) = L(n+1) + F(n) + Sum_{i=1.. n-2} F(i)*a(n-2-i), for F(n) = A000045(n) the Fibonacci numbers and L(n) = A000032(n) the Lucas numbers.
%F A354080 a(n) = L(n+1) + T(n+1) + Sum_{i=1.. n-2} L(i)*T(n-i), for L(n) = A000032(n) the Lucas numbers and T(n) = A000073(n) the tribonacci numbers.
%F A354080 G.f.: (1 + 3*x)/(1 - x - x^2 - x^3). - _Stefano Spezia_, Jul 14 2022
%e A354080 Here is one of the a(9)=392 tilings, this one with four squares, two dominoes, and one tromino.
%e A354080 _
%e A354080 |_|_______________
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%e A354080 |_|
%t A354080 LinearRecurrence[{1, 1, 1}, {1, 4, 5}, 50] (* _Paolo Xausa_, May 27 2024 *)
%Y A354080 Cf. A000032, A000045, A000073.
%K A354080 nonn,easy
%O A354080 0,2
%A A354080 _Greg Dresden_ and _Veda Garigipati_, Jul 13 2022