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

0,2

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).

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

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

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 *)

Crossrefs

Cf. A000930, A005178.

Sequence in context: A151220 A151221 A020515 * A010921 A275634 A163606

Adjacent sequences: A049083 A049084 A049085 * A049087 A049088 A049089

Keyword

easy,nonn

Author

Allan C. Wechsler

Extensions

More terms from Emeric Deutsch, Feb 15 2005

Status

approved