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

Author

Frank Ruskey, Sep 29 2010

Keywords

• nonn

Comments

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

Examples

			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
		

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).

Crossrefs

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

Programs

Magma

R:=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

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 *)

Sage

def A180970_list(prec):
    P. = 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

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).