A049086 Number of tilings of 4 X 3n rectangle by 1 X 3 rectangles. Rotations and reflections are considered distinct tilings.
1, 3, 13, 57, 249, 1087, 4745, 20713, 90417, 394691, 1722917, 7520929, 32830585, 143313055, 625594449, 2730863665, 11920848033, 52037243619, 227154537661, 991581805481, 4328482658041, 18894822411423, 82480245888473, 360045244866137, 1571680309076689, 6860746056673507
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
- R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 19.
- R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (10)
- Index entries for linear recurrences with constant coefficients, signature (5,-3,1).
Programs
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Maple
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 a := n -> hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -8/27): seq(simplify(a(n)), n=0..25); # Peter Luschny, Dec 09 2020
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Mathematica
LinearRecurrence[{5,-3,1},{1,3,13},50] (* Vincenzo Librandi, Feb 18 2012 *) CoefficientList[Series[(1-x)^2/(1-5x+3x^2-x^3), {x, 0, 40}], x] (* M. Poyraz Torcuk, Nov 06 2021 *)
Formula
a(n) = 5*a(n-1) - 3*a(n-2) + a(n-3).
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
G.f.: (1-x)^2/(1-5*x+3*x^2-x^3). - Colin Barker, Feb 03 2012
a(n) = Sum_{k=0..n} A109955(n,k)*2^k. - Philippe Deléham, Feb 18 2012
a(n) = hypergeom([(n+1)/2, n/2+1, -n], [1/3, 2/3], -8/27). - Peter Luschny, Dec 09 2020
Extensions
More terms from Emeric Deutsch, Feb 15 2005