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 A375821 #20 Sep 12 2024 11:30:38
%S A375821 1,1,2,7,17,41,107,274,693,1766,4504,11465,29194,74364,189391,482327,
%T A375821 1228412,3128559,7967841,20292639,51681711,131623900,335222103,
%U A375821 853749852,2174345752,5537663377,14103422348,35918853952,91478793557,232979863277,593357374262
%N A375821 Number of ways to tile a 3-row parallelogram of length n with triangular and rectangular tiles, each of size 3.
%C A375821 Here is the 3-row parallelogram of length 6 (with 18 cells):
%C A375821 ___ ___ ___ ___ ___ ___
%C A375821 | | | | | | |
%C A375821 _|___|___|___|___|_ _|___|
%C A375821 | | | | | | |
%C A375821 _|___|___|___|___|_ _|___|
%C A375821 | | | | | | |
%C A375821 |___|___|___|___|___|___|,
%C A375821 and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:
%C A375821 ___
%C A375821 | |
%C A375821 _|___|_ ___________
%C A375821 | | | | | | |
%C A375821 |___|___|, |___|___|___|.
%C A375821 As an example, here is one of the a(6) = 107 ways to tile the 3 x 6 parallelogram:
%C A375821 ___ _______ ___________
%C A375821 | | | |
%C A375821 _| _|_ _|___________|
%C A375821 | | | | |
%C A375821 _| _| |___|___________|
%C A375821 | | | |
%C A375821 |___|_______|___________|.
%H A375821 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,4,-1,0,-1).
%F A375821 a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).
%F A375821 G.f.: (1 - x - x^3)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).
%F A375821 a(n) = (A077939(n) + A077961(n))/2.
%t A375821 LinearRecurrence[{2, 0, 4, -1, 0, -1}, {1, 1, 2, 7, 17, 41}, 40]
%Y A375821 Cf. A077939, A077961, A375823.
%K A375821 nonn,easy
%O A375821 0,3
%A A375821 _Greg Dresden_ and Mingjun Oliver Ouyang, Aug 30 2024