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 A375823 #18 Sep 12 2024 15:09:49
%S A375823 0,1,3,6,16,43,107,271,695,1769,4499,11464,29202,74360,189382,482339,
%T A375823 1228417,3128538,7967848,20292665,51681683,131623881,335222157,
%U A375823 853749843,2174345679,5537663440,14103422412,35918853816,91478793556,232979863477,593357374127
%N A375823 Number of ways to tile a 3-row trapezoid of average length n with triangular and rectangular tiles, each of size 3.
%C A375823 Here is the 3-row trapezoid of average length 6 (with 18 cells):
%C A375823 ___ ___ ___ ___ ___
%C A375823 | | | | | |
%C A375823 _|___|___|___|___|_ _|_
%C A375823 | | | | | | |
%C A375823 _|___|___|___|___|_ _|___|_
%C A375823 | | | | | | | |
%C A375823 |___|___|___|___|___|___|___|,
%C A375823 and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:
%C A375823 ___
%C A375823 | |
%C A375823 _|___|_ ___________
%C A375823 | | | | | | |
%C A375823 |___|___|, |___|___|___|.
%C A375823 As an example, here is one of the a(6) = 107 ways to tile the 3-row trapezoid
%C A375823 ___ ___ ___________
%C A375823 | | | |
%C A375823 _| _|_ |___________|_
%C A375823 | | | | | |
%C A375823 _| _| |_ |_ _| |_
%C A375823 | | | | | |
%C A375823 |___|_______|___|___|_______|.
%H A375823 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,4,-1,0,-1).
%F A375823 a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).
%F A375823 G.f.: x*(1 + x)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).
%F A375823 a(n) = (A077939(n) - A077961(n))/2.
%t A375823 LinearRecurrence[{2, 0, 4, -1, 0, -1}, {0, 1, 3, 6, 16, 43}, 40]
%Y A375823 Cf. A077939, A077961, A375821.
%K A375823 nonn,easy
%O A375823 0,3
%A A375823 _Greg Dresden_ and Mingjun Oliver Ouyang, Aug 30 2024