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A180970 Number of tatami tilings of a 3 X n grid (with monomers allowed). 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 02:23 EST 2022. Contains 358572 sequences. (Running on oeis4.)