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A254124 The number of tilings of a 3 X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1.

%I A254124 #26 Dec 21 2022 08:19:43
%S A254124 1,4,29,257,2408,22873,217969,2078716,19827701,189133073,1804125632,
%T A254124 17209452337,164160078241,1565914710964,14937181915469,
%U A254124 142485030313697,1359157571347928,12964936038223753,123671875897903249,1179699833714208556,11253097663211943461
%N A254124 The number of tilings of a 3 X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1.
%C A254124 Let G_n be the graph with vertices {(a,b) : 1<=a<=5, 1<=b<=2n-1, a+b odd} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.  Then a(n) is the number of independent sets in G_n.
%H A254124 Colin Barker, <a href="/A254124/b254124.txt">Table of n, a(n) for n = 0..1000</a>
%H A254124 Z. Zhang, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match56/n3/match56n3_625-636.pdf">Merrifield-Simmons index of generalized Aztec diamond and related graphs</a>, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
%H A254124 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-24,5).
%F A254124 G.f.: (1 - 8*x + 5*x^2)/(1 - 12*x + 24*x^2 - 5*x^3).
%F A254124 a(n) = 12*a(n-1) - 24*a(n-2) + 5*a(n-3) for n > 2. - _Colin Barker_, Jun 07 2020
%o A254124 (PARI) Vec((1-8*x+5*x^2)/(1-12*x+24*x^2-5*x^3) + O(x^30)) \\ _Michel Marcus_, Jan 26 2015
%Y A254124 Cf. A052961, A254125, A254126, A254127.
%Y A254124 Column k=3 of A254414.
%K A254124 nonn,easy
%O A254124 0,2
%A A254124 _Steve Butler_, Jan 25 2015