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
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Feb 15 2005
STATUS
approved