A236579 The number of tilings of a 5 X (4n) floor with 1 X 4 tetrominoes.

1, 3, 15, 75, 371, 1833, 9057, 44753, 221137, 1092699, 5399327, 26679563, 131831075, 651413681, 3218814561, 15905050017, 78591236385, 388340962771, 1918899743823, 9481812581835, 46852249642771

Offset: 0

Author

R. J. Mathar, Jan 29 2014

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A003269 (4Xn floor), A236580 - A236582, A109960.

Programs

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 *)

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