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%I #29 Apr 06 2021 09:56:53
%S 1,3,13,22,44,90,196,406,852,1778,3740,7822,16404,34346,72004,150822,
%T 316076,662186,1387596,2907262,6091780,12763778,26744268,56036566,
%U 117413804,246015450,515476036,1080072022,2263070868,4741795442
%N Number of tatami tilings of a 3 X n grid (with monomers allowed).
%C A tatami tiling consists of dimers (1 X 2) and monomers (1 X 1) where no four meet at a point.
%D 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.
%H G. C. Greubel, <a href="/A180970/b180970.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, <a href="https://doi.org/10.37236/596">Monomer-Dimer Tatami Tilings of Rectangular Regions</a>, Electronic Journal of Combinatorics, 18(1) (2011) P109.
%H Alejandro Erickson, Frank Ruskey, Mark Schurch, and Jennifer Woodcock, <a href="https://arxiv.org/abs/1103.3309">Auspicious tatami mat arrangements</a>, arXiv:1103.3309 [math.CO], 2011. See p. 17.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,0,2,-1,-1).
%F 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).
%e 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:
%e hh hh hh hh hh hh vv vm vm mm mv mv mm
%e hh vv mv vm mm hh vv vv vm hh vv mv hh
%e hh vv mv vm hh mm hh mv hh hh vm hh mm
%t Join[{1,3,13}, LinearRecurrence[{1,2,0,2,-1,-1}, {22,44,90,196,406,852}, 37]] (* _Jean-François Alcover_, Jan 29 2019 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Integers(), 40);
%o 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
%o (Sage)
%o def A180970_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o 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()
%o A180970_list(40) # _G. C. Greubel_, Apr 05 2021
%Y Cf. A180965 (2 X n grid), A192090 (4 X n grid), row sums of A272472.
%K nonn
%O 0,2
%A _Frank Ruskey_, Sep 29 2010
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