A272471 Triangle T(n,m) by rows: The number of tatami tilings of a 2 X n grid with dimers and 2*m monomers.
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
Examples
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;
Links
- 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).
Programs
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Mathematica
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 *)
Formula
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