A272473 -id:A272473 - cp's OEIS frontend

A192090 Number of tatami tilings of a 4 X n grid (with monomers allowed).

1, 5, 29, 44, 66, 126, 238, 490, 922, 1714, 3306, 6246, 12102, 22994, 43682, 83810, 159154, 305062, 581382, 1108362, 2119602, 4037338, 7716554, 14720142, 28084702, 53639778, 102298794, 195341594, 372753634, 711338798, 1357975774

Offset: 0

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jun 23 2011

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 4 X 3 grid:
    _ _ _ _   _ _ _ _   _ _ _ _   _ _ _ _
   |_ _| |_| |_| |_ _| | |_ _| | |_| |_ _|
   |_ _|_| | | |_|_ _| |_| |_|_| | |_|_ _|
   |_|_ _|_| |_|_ _|_| |_|_|_ _| |_|_ _|_|

Alois P. Heinz, 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, 24 pages.

Cf. A180970, (3 X n grid), A192091 (5 X n grid), row sums of A272473.

G.f.: -13 + 3*x + 3*x^2 + 2*x^3 + (14 - 12*x + 10*x^2 + 10*x^4 - 104*x^5 + 114*x^6 - 80*x^7 + 34*x^8 + 12*x^9 - 2*x^10)/(1 - x - x^2 - x^3 + x^4 - 7*x^5 + 7*x^6 - x^7 + x^8 + x^9 + x^10 - x^11).

A068923 Number of ways to tile a 4 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

1, 4, 4, 2, 3, 3, 3, 5, 5, 6, 8, 8, 11, 13, 14, 19, 21, 25, 32, 35, 44, 53, 60, 76, 88, 104, 129, 148, 180, 217, 252, 309, 365, 432, 526, 617, 741, 891, 1049, 1267, 1508, 1790, 2158, 2557, 3057, 3666, 4347, 5215, 6223, 7404, 8881, 10570, 12619, 15104, 17974

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 3.
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1).

Cf. A068929 for incongruent tilings, A068920 for more info. First column of A272473.

For n >= 9, a(n) = a(n-3) + a(n-5).

G.f.: x*(x+1)*(2*x^6+x^5+x^4-x^2-3*x-1)/(-1+x^5+x^3) [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009]

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

A272474 Triangle T(n,m) by rows: the number of tatami tilings of the 5 X n floor with dimers and m monomers.

Original entry on oeis.org

0, 3, 0, 4, 0, 1, 6, 0, 35, 0, 26, 0, 1, 0, 18, 0, 56, 0, 16, 3, 0, 52, 0, 64, 0, 7, 0, 10, 0, 88, 0, 80, 2, 0, 60, 0, 182, 0, 81, 0, 8, 0, 160, 0, 320, 0, 96, 2, 0, 102, 0, 500, 0, 449, 0, 112, 0, 18, 0, 340, 0, 952, 0, 600, 0, 120, 4, 0, 184, 0, 1056

Offset: 1

R. J. Mathar, Apr 30 2016

			The triangle starts in row n=1 and column m=0 as:
0,3,0,4,0,1;
6,0,35,0,26,0,1;
0,18,0,56,0,16;
3,0,52,0,64,0,7;
0,10,0,88,0,80;
2,0,60,0,182,0,81;
0,8,0,160,0,320,0,96;
2,0,102,0,500,0,449,0,112;
0,18,0,340,0,952,0,600,0,120;
4,0,184,0,1056,0,1535,0,712,0,128;
0,24,0,550,0,2216,0,2338,0,824,0,128;
4,0,246,0,2050,0,4367,0,3256,0,936,0,128;
0,32,0,936,0,5044,0,7728,0,4454,0,1040,0,128;
6,0,414,0,4054,0,11539,0,12360,0,5816,0,1160,0,128;
0,52,0,1658,0,10736,0,22410,0,18744,0,7352,0,1280,0,128;
8,0,620,0,7412,0,27039,0,39590,0,26576,0,9056,0,1408,0,128;
0,68,0,2596,0,21180,0,57296,0,65634,0,36312,0,10864,0,1536,0,128;
10,0,908,0,13022,0,59625,0,112526,0,102054,0,47954,0,12816,0,1664,0,128;
0,100,0,4312,0,41056,0,138444,0,204496,0,152648,0,61720,0,14872,0,1792,0,128;
14,0,1404,0,23112,0,126291,0,298136,0,347122,0,219228,0,77904,0,17056,0,1920,0,128;
0,142,0,6904,0,77136,0,314464,0,584236,0,560856,0,305264,0,96552,0,19360,0,2048,0,128;
18,0,2034,0,38898,0,254427,0,731536,0,1068766,0,863460,0,413418,0,117944,0,21792,0,2176,0,128;
0,196,0,10778,0,139276,0,678728,0,1537620,0,1850598,0,1282412,0,546464,0,142128,0,24352,0,2304,0,128;
24,0,3018,0,65388,0,496213,0,1704232,0,3026128,0,3048168,0,1843736,0,707754,0,169288,0,27040,0,2432,0,128;

R. J. Mathar, Re: Tatami, Seqfan mailing list of Mar 24 2016.

Cf. A192091 (row sums), A068924 (column m=0), A281791 (column m=1), A272473 (4 by n grid).

G.f. x*( -3*x^11*y^8 +7*x^4*y^7 +x^2*y^7 -24*x^6*y^7 -2*x^5*y^4 -y^5 -2*x^3*y^2 -16*x^9 -9*x^ 2*y +3*x^3 +13*x^5 +7*x^15 -6*x -3*y -22*x*y^4 -32*x*y^2 +16*x^11*y^6 -6*x^13*y^2 -14*x^ 15*y^4 +3*x^13 +14*x^5*y^6 -11*x^11 +6*x^7 -55*y^3*x^10 -17*x^2*y^3 +37*x^6*y -24*y^3*x^14 -48*x^10*y +2*x^17*y^2 -19*x^12*y -4*y^3 -55*x^12*y^3 -8*x^11*y^4 +51*x^8*y^5 +28*x^7*y^6 +72*x^6*y^3 +31*y^2*x^15 -58*x^13*y^4 +x^17 -64*x^11*y^2 +36*x^10*y^5 +55*x^8*y^3 -50*x^6*y^5 -4*x^4*y^3 -5*x^9*y^4 -5*x^8*y +60*y^2*x^5 +84*x^7*y^2 -12*x^9*y^6 +55*x^7*y^4 -2*x^12*y^5 -79*x^9*y^2 -2*x^16*y^3 +4*x^7*y^8 +11*x^4*y -20*x^4*y^5 +6*x^13*y^ 6 +15*x^16*y +17*x^14*y^5 -5*x^10*y^7 -26*x^8*y^7 +x^8*y^9 +21*y*x^14 +11*x^2*y^5 +x^7* y^10 -x^10*y^9 +14*x^3*y^4 +9*x^3*y^6 -5*y^7*x^12 +3*x^9*y^8) / (x^14 +x^13*y +x^12 -2*x^ 12*y^2 -x^11*y^3 -2*x^10*y^2 -x^10 -x^9*y^3 +x^8*y^4 -3*x^8 -3*x^8*y^2 -x^7*y -x^7*y^3 +x^6*y^4 +4*x^6*y^2 -y^3*x^5 +2*x^4 +y^2*x^4 -x^3*y +x^2 +x*y -1). - R. J. Mathar, May 02 2016
G.f. for column m=1: x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. - R. J. Mathar, May 02 2016, corrected Apr 10 2017
G.f. for column m=2: -8 +17*x^2 +2*x^4 -2*(9*x^16 +24*x^14 +17*x^12 -22*x^10 -39*x^8 -9*x^6 +13*x^4 +9*x^2 +4) / (x^6+x^4-1)^3. - R. J. Mathar, May 02 2016

cp's OEIS Frontend

A192090 Number of tatami tilings of a 4 X n grid (with monomers allowed).

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A068923 Number of ways to tile a 4 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

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A272474 Triangle T(n,m) by rows: the number of tatami tilings of the 5 X n floor with dimers and m monomers.

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