OFFSET
0,2
COMMENTS
Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.
Related to A002378 by an Invert Transform.
LINKS
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 34.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], eq. (24).
Index entries for linear recurrences with constant coefficients, signature (6,-6,4,-1).
FORMULA
G.f.: (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4).
a(n) = Sum_{k = 0..n} binomial(n + 3*k, 4*k)*2^k = Sum_{k = 0..n} A109960(n,k)*2^k. - Peter Bala, Nov 02 2017
a(n) = hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128). - Peter Luschny, Nov 02 2017
MAPLE
g := (1-x)^3/(-6*x+1+6*x^2-4*x^3+x^4) ;
taylor(%, x=0, 30) ; gfun[seriestolist](%) ;
# Alternatively:
a := n -> hypergeom([(n+1)/3, (n+2)/3, n/3 + 1, -n], [1/4, 1/2, 3/4], -27/128):
seq(simplify(a(n)), n=0..20); # Peter Luschny, Nov 02 2017
MATHEMATICA
LinearRecurrence[{6, -6, 4, -1}, {1, 3, 15, 75}, 21] (* Jean-François Alcover, Jul 14 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jan 29 2014
STATUS
approved