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%I #54 Dec 10 2022 09:29:17
%S 1,3,13,57,249,1087,4745,20713,90417,394691,1722917,7520929,32830585,
%T 143313055,625594449,2730863665,11920848033,52037243619,227154537661,
%U 991581805481,4328482658041,18894822411423,82480245888473,360045244866137,1571680309076689,6860746056673507
%N Number of tilings of 4 X 3n rectangle by 1 X 3 rectangles. Rotations and reflections are considered distinct tilings.
%H Vincenzo Librandi, <a href="/A049086/b049086.txt">Table of n, a(n) for n = 0..1000</a>
%H Mudit Aggarwal and Samrith Ram, <a href="https://arxiv.org/abs/2206.04437">Generating functions for straight polyomino tilings of narrow rectangles</a>, arXiv:2206.04437 [math.CO], 2022.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings</a>, arXiv:1311.6135 [math.CO], 2013, Table 19.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1406.7788">Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices</a>, arXiv:1406.7788 (2014), eq. (10)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,1).
%F a(n) = 5*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n)/a(n-1) tends to 4.3652300134..., an eigenvalue of the matrix M and an inverse root of the polynomial x^3 - 3x^2 + 5x - 1. [a(n-2), a(n-1), a(n)] = M^n * [1 1 1], where M = the 3 X 3 matrix [ 5 -3 1 / 1 0 0 / 0 1 0]. E.g., a(3), a(4), a(5) = 57, 249, 1087. M^5 * [1 1 1] = [57, 249, 1087] - _Gary W. Adamson_, Apr 25 2004
%F G.f.: (1-x)^2/(1-5*x+3*x^2-x^3). - _Colin Barker_, Feb 03 2012
%F a(n) = Sum_{k=0..n} A109955(n,k)*2^k. - _Philippe Deléham_, Feb 18 2012
%F a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -8/27). - _Peter Luschny_, Dec 09 2020
%p a[0]:=1:a[1]:=3:a[2]:=13: for n from 3 to 25 do a[n]:=5*a[n-1]-3*a[n-2]+a[n-3] od: seq(a[n],n=0..25); # _Emeric Deutsch_, Feb 15 2005
%p a := n -> hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -8/27):
%p seq(simplify(a(n)), n=0..25); # _Peter Luschny_, Dec 09 2020
%t LinearRecurrence[{5,-3,1},{1,3,13},50] (* _Vincenzo Librandi_, Feb 18 2012 *)
%t CoefficientList[Series[(1-x)^2/(1-5x+3x^2-x^3), {x, 0, 40}], x] (* _M. Poyraz Torcuk_, Nov 06 2021 *)
%Y Cf. A000930, A005178.
%K easy,nonn
%O 0,2
%A _Allan C. Wechsler_
%E More terms from _Emeric Deutsch_, Feb 15 2005
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