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
KEYWORD
nonn
AUTHOR
Frank Ruskey, Sep 29 2010
STATUS
approved