A180965 -id:A180965 - 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).

A272471 Triangle T(n,m) by rows: The number of tatami tilings of a 2 X n grid with dimers and 2*m monomers.

Original entry on oeis.org

1, 1, 2, 4, 3, 9, 1, 4, 18, 7, 6, 35, 26, 1, 9, 64, 73, 10, 13, 112, 179, 52, 1, 19, 192, 403, 194, 13, 28, 323, 850, 597, 87, 1, 41, 534, 1707, 1624, 408, 16, 60, 872, 3303, 4046, 1532, 131, 1, 88, 1410, 6203, 9428, 4951, 742, 19, 129, 2260, 11366, 20847, 14361, 3308

Offset: 1

R. J. Mathar, Apr 30 2016

			The triangle starts in row n=1 as:
1,1;
2,4;
3,9,1;
4,18,7;
6,35,26,1;
9,64,73,10;
13,112,179,52,1;
19,192,403,194,13;
28,323,850,597,87,1;
41,534,1707,1624,408,16;
60,872,3303,4046,1532,131,1;
88,1410,6203,9428,4951,742,19;
129,2260,11366,20847,14361,3308,184;
189,3596,20407,44194,38369,12472,1223,22;
277,5687,36018,90492,96071,41559,6330,246,1;
406,8946,62648,179982,228224,125942,27382,1878,25;
595,14007,107602,349244,519071,353929,103504,11084,317,1;
872,21842,182800,663470,1138094,935298,352234,54226,2734,28;
1278,33937,307581,1237436,2418496,2348345,1101887,229886,18137,397,1;
1873,52560,513165,2271066,5002459,5646544,3216889,870490,99142,3818,31;
2745,81168,849726,4109303,10106351,13082702,8864264,3008357,466676,28137,486,1;
4023,125022,1397565,7342252,19999147,29352862,23256181,9640660,1949989,170104,5157,34;
5896,192117,2284716,12971722,38856275,64030031,58492346,28994720,7393154,881310,41813,584,1;
8641,294588,3714618,22686720,74268850,136248856,141811450,82607092,25852870,4033500,277165,6778,37;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
A. Erickson, F. Ruskey, J. Woodcock, M. Schurch, Auspicious tatami mat arrangements, arXiv:1103.3309 (2011).

Cf. A180965 (row sums), A000930 (column m=0), A046741 (without tatami condition).

Select[Flatten[CoefficientList[CoefficientList[Series[x*(y^2 + 2*x*y^2 + x^2*y^2 + 1 - x^3)/(x^4 - x^3*y^2 - x^3 - x^2*y^2 + x^2 - 2*x + 1), {x, 0, 10}, {y, 0, 10}], x], y]], # != 0 &] (* G. C. Greubel, Apr 28 2017 *)

G.f. x*(y^2 +2*x*y^2 +x^2*y^2 +1 -x^3)/(x^4 -x^3*y^2 -x^3 -x^2*y^2 +x^2 -2*x +1). - R. J. Mathar, May 01 2016

cp's OEIS Frontend

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

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