A180970 Number of tatami tilings of a 3 X n grid (with monomers allowed).

1, 3, 13, 22, 44, 90, 196, 406, 852, 1778, 3740, 7822, 16404, 34346, 72004, 150822, 316076, 662186, 1387596, 2907262, 6091780, 12763778, 26744268, 56036566, 117413804, 246015450, 515476036, 1080072022, 2263070868, 4741795442

Offset

0,2

Comments

A tatami tiling consists of dimers (1 X 2) and monomers (1 X 1) where no four meet at a point.

References

A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Auspicious Tatami Mat Arrangements, The 16th Annual International Computing and Combinatorics Conference (COCOON 2010), July 19-21, Nha Trang, Vietnam. LNCS 6196 (2010) 288-297.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Monomer-Dimer Tatami Tilings of Rectangular Regions, Electronic Journal of Combinatorics, 18(1) (2011) P109.

Alejandro Erickson, Frank Ruskey, Mark Schurch, and Jennifer Woodcock, Auspicious tatami mat arrangements, arXiv:1103.3309 [math.CO], 2011. See p. 17.

Index entries for linear recurrences with constant coefficients, signature (1,2,0,2,-1,-1).

Formula

G.f.: (1 + 2*x + 8*x^2 + 3*x^3 - 6*x^4 - 3*x^5 - 4*x^6 + 2*x^7 + x^8)/(1 - x - 2*x^2 - 2*x^4 + x^5 + x^6).

Example

Below we show the a(2) = 13 tatami tilings of a 2 X 3 rectangle where v = square of a vertical dimer, h = square of a horizontal dimer, m = monomer:
hh hh hh hh hh hh vv vm vm mm mv mv mm
hh vv mv vm mm hh vv vv vm hh vv mv hh
hh vv mv vm hh mm hh mv hh hh vm hh mm

Mathematica

Join[{1, 3, 13}, LinearRecurrence[{1, 2, 0, 2, -1, -1}, {22, 44, 90, 196, 406, 852}, 37]] (* Jean-François Alcover, Jan 29 2019 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1 +2*x +8*x^2 +3*x^3 -6*x^4 -3*x^5 -4*x^6 +2*x^7 +x^8)/(1 -x -2*x^2 -2*x^4 +x^5 +x^6) )); // G. C. Greubel, Apr 05 2021
(Sage)
def A180970_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1 +2*x +8*x^2 +3*x^3 -6*x^4 -3*x^5 -4*x^6 +2*x^7 +x^8)/(1 -x -2*x^2 -2*x^4 +x^5 +x^6) ).list()
A180970_list(40) # G. C. Greubel, Apr 05 2021

Crossrefs

Cf. A180965 (2 X n grid), A192090 (4 X n grid), row sums of A272472.

Sequence in context: A258774 A057589 A299645 * A135580 A166566 A011533

Adjacent sequences: A180967 A180968 A180969 * A180971 A180972 A180973

Keyword

nonn

Author

Frank Ruskey, Sep 29 2010

Status

approved