A192090 -id:A192090 - cp's OEIS frontend

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

Frank Ruskey, Sep 29 2010

nonn

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

			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

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.

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).

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

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

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

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

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).

A192091 Number of tatami tilings of a 5 X n grid (with monomers allowed).

Original entry on oeis.org

1, 8, 68, 90, 126, 178, 325, 584, 1165, 2030, 3619, 6080, 10987, 19362, 35477, 62360, 111837, 195614, 350707, 619568, 1112315, 1967090, 3514597, 6214984, 11093549, 19664558, 35090115, 62247552, 110934699, 196859394, 350650261

Offset: 0

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 07 2011

nonn

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

			Here are some tatami tilings of the 5 X 3 grid:
    _ _ _ _ _   _ _ _ _ _   _ _ _ _ _   _ _ _ _ _
   |_ _| |_| | |_| |_|_ _| | |_ _| |_| |_| | |_ _|
   |_ _|_| |_| | |_|_ _| | |_| |_|_| | | |_|_| | |
   |_|_ _|_|_| |_|_ _|_|_| |_|_|_ _|_| |_|_ _|_|_|

A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Monomer-Dimer Tatami Tilings of Rectangular Regions, Electronic Journal of Combinatorics, 18(1) (2011) P109, 24 pages.

Cf. A192090, A192092, A033508 (without tatami condition). Row sums of A272474.

A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

Original entry on oeis.org

1, 3, 1, 4, 18, 7, 4, 27, 13, 2, 32, 32, 3, 52, 64, 7, 3, 62, 133, 40, 3, 99, 269, 110, 9, 5, 152, 437, 280, 48, 5, 163, 730, 669, 138, 9, 6, 258, 1243, 1318, 433, 48, 8, 343, 1823, 2670, 1239, 154, 9, 8, 408, 2949, 5240, 2849, 600, 48, 11, 632, 4577

Offset: 1

R. J. Mathar, Apr 30 2016

The number of squares in the 4 by n floor is even, so the number of tilings with an odd number of monomers is zero.

			The triangle starts in row n=1 and column m=0 as:
1,3,1;
4,18,7;
4,27,13;
2,32,32;
3,52,64,7;
3,62,133,40;
3,99,269,110,9;
5,152,437,280,48;
5,163,730,669,138,9;
6,258,1243,1318,433,48;
8,343,1823,2670,1239,154,9;
8,408,2949,5240,2849,600,48;
11,632,4577,9011,6655,1927,172,9;
13,746,6287,16184,14697,4930,777,48;
14,971,9928,28135,28805,13089,2669,190,9;
19,1394,14234,44806,58022,32176,7501,954,48;
21,1610,19501,75702,111795,70427,22344,3445,208,9;
25,2224,29785,121302,199354,157078,59859,10576,1131,48;
32,2909,40073,184597,366553,331449,143611,34646,4257,226,9;
35,3464,55939,298278,644436,651772,350855,99300,14167,1308,48;
44,4820,81474,449995,1081033,1303651,802565,258303,50095,5105,244,9;
53,5924,106460,670726,1868914,2488996,1719501,684338,151835,18274,1485,48;
60,7408,150672,1040424,3077401,4548409,3716945,1678785,425017,68761,5989,262,9;
76,9972,208211,1503372,4956628,8434302,7641320,3879356,1208052,218806,22897,1662,48;

Cf. A192090 (row sums), A068923 (column m=0), A272472 (3 by n grid), A100265 (without tatami condition, reversed rows).

G.f. x*( -1 -8*x^7*y^2 +21*x^5*y^2 -7*x^7*y^6 +4*x^3*y^2 -3*x^7 +2*x^5 -8*x^2*y^2 -4*x^8*y^4 -3*x -6*x*y^4 -15*x*y^2 -2*x^3*y^4 -6*x^8 -5*x^10*y^2 -5*x^9*y^2 -y^4 -2*x^8*y^2 -3*y^2 -8*x^11*y^2 +5*x^11*y^4 -3*x^2*y^4 -2*x^5*y^6 +2*x^13 +x^12 +x^11 +x^6 -7*x^7*y^4 +x^7*y^8 +11*x^4*y^2 -3*x^9 -15*x^10*y^4 -2*x^10*y^6 +18*x^9*y^4 +36*x^6*y^4 +20*x^6*y^2 -17*x^ 5*y^4 -8*x^4*y^4 +4*x^3 +8*x^6*y^6 +5*x^4 +2*x^9*y^6 -y^8*x^6 +6*y^6*x^3 +y^6*x^2)/ (x^11 -x^10 +2*x^9 -3*x^9*y^2 +x^8*y^2 -2*x^8 +x^7 +x^6*y^4 -5*x^6*y^2 -3*x^6 +2*x^5 +5*x^5*y^2 +x^4*y^2 -2*x^4 -x^3*y^2 +2*x^3 +x^2*y^2 +x -1). - R. J. Mathar, May 01 2016

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

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A192091 Number of tatami tilings of a 5 X n grid (with monomers allowed).

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A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

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