A236577 The number of tilings of a 6 X n floor with 1 X 3 trominoes.

1, 1, 1, 6, 13, 22, 64, 155, 321, 783, 1888, 4233, 9912, 23494, 54177, 126019, 295681, 687690, 1600185, 3738332, 8712992, 20293761, 47337405, 110368563, 257206012, 599684007, 1398149988, 3259051800, 7597720649, 17712981963

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.

Links

• G. C. Greubel, 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 21.
• R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq (14).
• Index entries for linear recurrences with constant coefficients, signature (1,1,7,-1,-5,-10,-1,3,5,1,-1,-1).

Crossrefs

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576 - A236578.

Column k=3 of A250662.

Cf. A251073.

Programs

Maple

g := (1-x^3)^2*(-x^2+1-x^3)/ (-x^10+x^12+x^11+10*x^6-5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

Mathematica

CoefficientList[Series[(1 - x^3)^2*(-x^2 + 1 - x^3)/(-x^10 + x^12 + x^11 + 10*x^6 - 5*x^9 - 3*x^8 + x^7 + x^4 - 7*x^3 + 5*x^5 - x^2 - x + 1), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)

PARI

x='x+O('x^50); Vec((1-x^3)^2*(-x^2+1-x^3)/(-x^10+x^12+x^11+10*x^6 -5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1)) \\ G. C. Greubel, Apr 27 2017

Formula

G.f.: See the definition of g in the Maple code.