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A049086 Number of tilings of 4 X 3n rectangle by 1 X 3 rectangles. Rotations and reflections are considered distinct tilings. 5
1, 3, 13, 57, 249, 1087, 4745, 20713, 90417, 394691, 1722917, 7520929, 32830585, 143313055, 625594449, 2730863665, 11920848033, 52037243619, 227154537661, 991581805481, 4328482658041, 18894822411423, 82480245888473 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The inverse INVERT transform yields A052530 (where A052530(0)=1). - R. J. Mathar, Nov 22 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 19.

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

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

MATHEMATICA

LinearRecurrence[{5, -3, 1}, {1, 3, 13}, 50] (* Vincenzo Librandi, Feb 18 2012 *)

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

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Last modified September 18 12:28 EDT 2020. Contains 337169 sequences. (Running on oeis4.)