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%I #47 Sep 08 2022 08:46:25
%S 3,2,4,11,24,52,115,254,560,1235,2724,6008,13251,29226,64460,142171,
%T 313568,691596,1525363,3364294,7420184,16365731,36095756,79611696,
%U 175589123,387274002,854159700,1883908523,4155091048,9164341796,20212592115,44580275278,98324892352
%N a(n) = 2*a(n-1) + a(n-3) with a(0) = 3, a(1) = 2, a(2) = 4.
%C a(n) is the number of ways to tile a bracelet of length n with black trominos, and black or white squares.
%H Colin Barker, <a href="/A332647/b332647.txt">Table of n, a(n) for n = 0..1000</a>
%H Greg Dresden and Michael Tulskikh, <a href="https://dresden.academic.wlu.edu/files/2021/01/TwoByNWeb.pdf">Tilings of 2 X n boards with dominos and L-shaped trominos</a>, Washington & Lee University (2021).
%H Helmut Prodinger, <a href="https://arxiv.org/abs/2011.04388">On third-order Pell polynomials</a>, arXiv:2011.04388 [math.NT], 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1).
%F a(n) = 2*a(n-1) + a(n-3).
%F a(n) = w1^n + w2^n + w3^n where w1,w2,w3 are the three roots of x^3-2x^2-1=0.
%F For n>2, a(n) = round(w1^n) for w1 the single real root of x^3-2x^2-1=0.
%F G.f.: (3 - 4*x) / (1 - 2*x - x^3). - _Colin Barker_, Feb 18 2020
%F a(n) = A008998(n) + 2*A008998(n-3) = 3*A008998(n) - 4*A008998(n-1).
%F a(n) = (5*b(n) - b(n-1) - b(n-2))/2 where b(n) = A052980(n). - _Greg Dresden_, Mar 10 2020
%F a(n) = A080204(n) + 1. - _Greg Dresden_, May 27 2020
%t LinearRecurrence[{2, 0, 1}, {3, 2, 4}, 50]
%o (Magma) a:=[3,2,4]; [n le 3 select a[n] else 2*Self(n-1)+Self(n-3):n in [1..33]]; // _Marius A. Burtea_, Feb 18 2020
%o (PARI) Vec((3 - 4*x) / (1 - 2*x - x^3) + O(x^30)) \\ _Colin Barker_, Feb 18 2020
%o (PARI) polsym(x^3-2*x^2-1, 44) \\ _Joerg Arndt_, May 28 2020
%Y Cf. A008998, A052980. Equals one more than A080204.
%K easy,nonn
%O 0,1
%A _Greg Dresden_, Feb 18 2020
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