POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

Then click "Show" button.

You may also see list of all polyomino sets for which data is available here.


U pentomino and X pentomino

Prime rectangles: ≥ 0.

Smallest rectangle tilings

Smallest rectangle (3x5):

Smallest square (15x15):

Rectangle tilings' solutions count (including symmetric)

Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Purple number - prime rectangle (unknown if weakly or strongly prime).

Red number - composite rectangle (which can be divided into two rectangles tileable by this set).

Gray number - it is unknown whether rectangle is prime or composite.

Question mark (?) - solution count is unknown.

Click on underlined numbers to view picture with one solution.

w \ h
1-2
3
4
5
6
7
8
9
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11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
N>0
1-2
0
3
0
0
4
0
0
0
5
0
1
0
0
6
0
0
0
1
0
7
0
0
0
0
0
0
8
0
0
0
0
0
0
0
9
0
0
0
1
0
0
0
0
10
0
1
0
0
1
0
0
1
0
11
0
0
0
0
0
0
0
0
0
0
12
0
0
0
1
0
0
0
0
1
0
0
13
0
0
0
0
0
0
0
0
0
0
0
0
14
0
0
0
0
0
0
0
0
0
0
0
0
0
15
0
1
0
1
1
0
2
1
1
3
1
3
4
2
16
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
17
0
0
0
0
0
0
0
0
0
0
0
0
0
5
0
0
18
0
0
0
1
0
0
0
0
1
0
0
0
0
5
0
0
0
19
0
0
0
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
20
0
1
0
0
1
0
0
1
0
0
1
0
0
7
0
0
17
0
0
21
0
0
0
1
0
0
0
0
1
0
0
0
0
11
0
0
0
0
33
0
22
0
0
0
0
0
0
0
0
0
0
0
0
0
15
0
0
0
0
0
0
0
23
0
0
0
0
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
0
24
0
0
0
1
0
0
0
0
1
0
0
0
0
21
0
0
0
0
57
0
0
0
0
25
0
1
0
0
1
0
0
1
0
0
1
0
0
22
0
0
49
0
0
85
?
?
≥1
?
26
0
0
0
0
0
0
0
0
0
0
0
0
0
23
0
0
0
0
0
0
0
0
0
?
0
27
0
0
0
1
0
0
0
0
1
0
0
0
0
36
0
0
0
0
91
0
0
0
0
≥1
0
0
28
0
0
0
0
0
0
0
0
0
0
0
0
0
34
0
0
0
0
0
0
0
0
0
≥1
0
0
0
29
0
0
0
0
0
0
0
0
0
0
0
0
0
44
0
0
0
0
0
0
0
0
0
?
0
0
0
?
30
0
1
0
1
1
0
4
1
1
9
1
9
16
58
36
25
137
100
173
297
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
0
0
0
0
0
0
0
0
0
0
0
0
0
57
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
32
0
0
0
0
0
0
0
0
0
0
0
0
0
80
0
0
0
0
0
0
0
0
0
?
0
0
0
?
33
0
0
0
1
0
0
0
0
1
0
0
0
0
92
0
0
0
0
355
0
0
0
0
≥1
0
0
0
?
34
0
0
0
0
0
0
0
0
0
0
0
0
0
101
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
35
0
1
0
0
1
0
0
1
0
0
1
0
0
138
0
0
375
0
0
954
?
≥1
≥1
?
≥1
≥1
≥1
?
36
0
0
0
1
0
0
0
0
1
0
0
0
0
149
0
0
0
0
725
0
0
0
0
≥1
0
0
0
?
37
0
0
0
0
0
0
0
0
0
0
0
0
0
181
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
38
0
0
0
0
0
0
0
0
0
0
0
0
0
230
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
39
0
0
0
1
0
0
0
0
1
0
0
0
0
250
0
0
0
0
1429
0
0
0
0
≥1
0
0
0
?
40
0
1
0
0
1
0
0
1
0
0
1
0
0
319
0
0
1029
8
0
2863
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
41
0
0
0
0
0
0
0
0
0
0
0
0
0
379
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
42
0
0
0
1
0
0
0
0
1
0
0
0
0
431
0
0
0
0
2689
0
0
0
0
≥1
0
0
0
?
43
0
0
0
0
0
0
0
0
0
0
0
0
0
549
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
44
0
0
0
0
0
0
0
0
0
0
0
0
0
629
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
45
0
1
0
1
1
0
8
1
1
27
1
27
64
750
216
125
2859
1000
5043
9028
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
46
0
0
0
0
0
0
0
0
0
0
0
0
0
928
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
47
0
0
0
0
0
0
0
0
0
0
0
0
0
1060
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
48
0
0
0
1
0
0
0
0
1
0
0
0
0
1299
0
0
0
0
9613
0
0
0
0
≥1
0
0
0
?
49
0
0
0
0
0
0
0
0
0
0
0
0
0
1557
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
50
0
1
0
0
1
0
0
1
0
0
1
0
0
1810
0
0
7985
0
0
28599
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
51
0
0
0
1
0
0
0
0
1
0
0
0
0
2227
0
0
0
0
18609
0
0
0
0
≥1
0
0
0
?
52
0
0
0
0
0
0
0
0
0
0
0
0
0
2617
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
53
0
0
0
0
0
0
0
0
0
0
0
0
0
3109
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
54
0
0
0
1
0
0
0
0
1
0
0
0
0
3784
0
0
0
0
36237
0
0
0
0
≥1
0
0
0
?
55
0
1
0
0
1
0
0
1
0
0
1
0
0
4427
0
0
22289
224
0
89133
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
56
0
0
0
0
0
0
0
0
0
0
0
0
0
5336
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
57
0
0
0
1
0
0
0
0
1
0
0
0
0
6401
0
0
0
0
70115
0
0
0
0
≥1
0
0
0
?
58
0
0
0
0
0
0
0
0
0
0
0
0
0
7536
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
59
0
0
0
0
0
0
0
0
0
0
0
0
0
9120
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
60
0
1
0
1
1
0
16
1
1
81
1
81
256
10828
1296
627
62121
10000
134729
279181
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
61
0
0
0
0
0
0
0
0
0
0
0
0
0
12872
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
62
0
0
0
0
0
0
0
0
0
0
0
0
0
15521
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
63
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
18364
0
0
0
0
258261
0
0
0
0
≥1
0
0
0
?
64
0
0
0
0
0
0
0
0
0
0
0
0
0
21992
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
65
0
1
0
0
≥1
0
0
1
0
0
1
0
0
26349
0
0
172989
32
0
877835
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
66
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
31236
0
0
0
0
495835
0
0
0
0
≥1
0
0
0
?
67
0
0
0
0
0
0
0
0
0
0
0
0
0
37513
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
68
0
0
0
0
0
0
0
0
0
0
0
0
0
44713
0
0
0
0
0
0
0
0
0
≥1
0
0
0
?
69
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
53228
0
0
0
0
954827
0
0
0
0
≥1
0
0
0
?
70
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
63862
0
0
481661
4450
2
2754490
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
71
0
0
0
0
0
0
0
0
0
0
0
0
0
75949
0
0
0
0
2
0
0
0
0
≥1
0
0
0
?
72
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
90741
0
0
0
0
1841135
0
0
0
0
≥1
0
0
0
?
73
0
0
0
0
0
0
0
0
0
0
0
0
0
108575
0
0
0
0
8
0
0
0
0
≥1
0
0
0
?
74
0
0
0
0
0
0
0
0
0
0
0
0
0
129177
0
0
0
0
20
0
0
0
0
≥1
0
0
0
?
75
0
1
0
≥1
≥1
0
32
≥1
≥1
243
1
243
1024
154603
7776
3153
1341343
100000
3548313
8647825
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
76
0
0
0
0
0
0
0
0
0
0
0
0
0
184524
0
0
0
0
14
0
0
0
0
≥1
0
0
0
?
77
0
0
0
0
0
0
0
0
0
0
0
0
0
219918
0
0
0
0
38
0
0
0
0
≥1
0
0
0
?
78
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
263178
0
0
0
0
6830743
0
0
0
0
≥1
0
0
0
?
79
0
0
0
0
0
0
0
0
0
0
0
0
0
313701
0
0
0
0
28
0
0
0
0
≥1
0
0
0
?
80
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
374521
0
0
3735973
704
62
27187329
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
81
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
447702
0
0
0
0
13140755
0
0
0
0
≥1
0
0
0
?
82
0
0
0
0
0
0
0
0
0
0
0
0
0
533619
0
0
0
0
44
0
0
0
0
≥1
0
0
0
?
83
0
0
0
0
0
0
0
0
0
0
0
0
0
637699
0
0
0
0
90
0
0
0
0
≥1
0
0
0
?
84
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
761403
0
0
0
0
25282555
0
0
0
0
≥1
0
0
0
?
85
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
908140
0
38
10406063
74032
180
85480748
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
86
0
0
0
0
0
0
0
0
0
0
0
0
0
1085401
0
0
0
0
154
0
0
0
0
≥1
0
0
0
?
87
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1295022
0
0
0
0
48665023
0
0
0
0
≥1
0
0
0
?
88
0
0
0
0
0
0
0
0
0
0
0
0
0
1545839
0
0
0
0
424
0
0
0
0
≥1
0
0
0
?
89
0
0
0
0
0
0
0
0
0
0
0
0
0
1846804
0
0
0
0
686
0
0
0
0
≥1
0
0
0
?
90
0
1
0
≥1
≥1
0
64
≥1
≥1
729
1
729
4096
2203162
46656
15891
28984261
1000256
93701113
268795742
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
91
0
0
0
0
0
0
0
0
0
0
0
0
0
2631240
0
0
0
0
832
0
0
0
0
≥1
0
0
0
?
92
0
0
0
0
0
0
0
0
0
0
0
0
0
3141826
0
0
0
0
1620
0
0
0
0
≥1
0
0
0
?
93
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
3749001
0
0
0
0
180415211
0
0
0
0
≥1
0
0
0
?
94
0
0
0
0
0
0
0
0
0
0
0
0
0
4478044
0
0
0
0
1664
0
0
0
0
≥1
0
0
0
?
95
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
5344988
0
0
80728761
12560
3328
845533550
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
96
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
6380241
0
0
0
0
347318287
0
0
0
0
≥1
0
0
0
?
97
0
0
0
0
0
0
0
0
0
0
0
0
0
7619870
0
0
0
0
3052
0
0
0
0
≥1
0
0
0
?
98
0
0
0
0
0
0
0
0
0
0
0
0
0
9093989
0
0
0
0
6234
0
0
0
0
≥1
0
0
0
?
99
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
10858285
0
0
0
0
668532591
0
0
0
0
≥1
0
0
0
?
100
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
12964858
0
778
224848577
1101586
7656
2.66019017×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
101
0
0
0
0
0
0
0
0
0
0
0
0
0
15474230
0
0
0
0
11292
0
0
0
0
≥1
0
0
0
?
102
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
18478155
0
0
0
0
1.28679727×10¹⁰
0
0
0
0
≥1
0
0
0
?
103
0
0
0
0
0
0
0
0
0
0
0
0
0
22058847
0
0
0
0
16878
0
0
0
0
≥1
0
0
0
?
104
0
0
0
0
0
0
0
0
0
0
0
0
0
26332515
0
0
0
0
26324
0
0
0
0
≥1
0
0
0
?
105
0
1
0
≥1
≥1
0
128
≥1
≥1
2187
1
2187
16384
31443013
279936
80259
626256337
10010112
2.47699042×10¹⁰
8.37032606×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
106
0
0
0
0
0
0
0
0
0
0
0
0
0
37533077
0
0
0
0
35156
0
0
0
0
≥1
0
0
0
?
107
0
0
0
0
0
0
0
0
0
0
0
0
0
44810670
0
0
0
0
57710
0
0
0
0
≥1
0
0
0
?
108
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
53501860
0
0
0
0
4.76830889×10¹⁰
0
0
0
0
≥1
0
0
0
?
109
0
0
0
0
0
0
0
0
0
0
0
0
0
63865592
0
0
0
0
71314
0
0
0
0
≥1
0
0
0
?
110
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
76253683
0
20
1.74427689×10¹⁰
214592
122196
2.63414861×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
111
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
91034937
0
0
0
0
9.17932380×10¹⁰
0
0
0
0
≥1
0
0
0
?
112
0
0
0
0
0
0
0
0
0
0
0
0
0
108676262
0
0
0
0
139124
0
0
0
0
≥1
0
0
0
?
113
0
0
0
0
0
0
0
0
0
0
0
0
0
129755543
0
0
0
0
248072
0
0
0
0
≥1
0
0
0
?
114
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
154900529
0
0
0
0
1.76704540×10¹¹
0
0
0
0
≥1
0
0
0
?
115
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
184929945
0
9646
4.85824613×10¹⁰
15190456
294508
8.29082702×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
116
0
0
0
0
0
0
0
0
0
0
0
0
0
220790480
0
0
0
0
487712
0
0
0
0
≥1
0
0
0
?
117
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
263576791
0
0
0
0
3.40152623×10¹¹
0
0
0
0
≥1
0
0
0
?
118
0
0
0
0
0
0
0
0
0
0
0
0
0
314685488
0
0
0
0
613328
0
0
0
0
≥1
0
0
0
?
119
0
0
0
0
0
0
0
0
0
0
0
0
0
375691009
0
0
0
0
1003354
0
0
0
0
≥1
0
0
0
?
120
0
1
0
≥1
≥1
0
256
≥1
≥1
6561
1
6561
65536
448506736
1679616
406185
1.35314308×10¹¹
100276480
6.54781103×10¹¹
2.60978451×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
121
0
0
0
0
0
0
0
0
0
0
0
0
0
535475968
0
0
0
0
1268892
0
0
0
0
≥1
0
0
0
?
122
0
0
0
0
0
0
0
0
0
0
0
0
0
639267800
0
0
0
0
2061402
0
0
0
0
≥1
0
0
0
?
123
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
763192224
0
0
0
0
1.26043783×10¹²
0
0
0
0
≥1
0
0
0
?
124
0
0
0
0
0
0
0
0
0
0
0
0
0
911166977
0
0
0
0
2585102
0
0
0
0
≥1
0
0
0
?
125
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
1.08777453×10¹⁰
0
882
3.76884070×10¹¹
3604946
4230936
8.21599466×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
126
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1.29866819×10¹⁰
0
0
0
0
2.42633782×10¹²
0
0
0
0
≥1
0
0
0
?
127
0
0
0
0
0
0
0
0
0
0
0
0
0
1.55043477×10¹⁰
0
0
0
0
5169428
0
0
0
0
≥1
0
0
0
?
128
0
0
0
0
0
0
0
0
0
0
0
0
0
1.85096676×10¹⁰
0
0
0
0
8597296
0
0
0
0
≥1
0
0
0
?
129
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
2.20983516×10¹⁰
0
0
0
0
4.67071245×10¹²
0
0
0
0
≥1
0
0
0
?
130
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
2.63820931×10¹⁰
0
94100
1.04971574×10¹²
198370314
10499286
2.58678991×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
131
0
0
0
0
0
0
0
0
0
0
0
0
0
3.14963495×10¹⁰
0
0
0
0
17268170
0
0
0
0
≥1
0
0
0
?
132
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
3.76026994×10¹⁰
0
0
0
0
8.99112730×10¹²
0
0
0
0
≥1
0
0
0
?
133
0
0
0
0
0
0
0
0
0
0
0
0
0
4.48917607×10¹⁰
0
0
0
0
21243368
0
0
0
0
≥1
0
0
0
?
134
0
0
0
0
0
0
0
0
0
0
0
0
0
5.35947012×10¹⁰
0
0
0
0
34807644
0
0
0
0
≥1
0
0
0
?
135
0
1
0
≥1
≥1
0
512
≥1
≥1
19683
1
19683
262144
6.39847925×10¹⁰
10077696
2059763
2.92371878×10¹²
1.00606851×10¹⁰
1.73078576×10¹³
8.14521863×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
136
0
0
0
0
0
0
0
0
0
0
0
0
0
7.63881102×10¹⁰
0
0
0
0
43060770
0
0
0
0
≥1
0
0
0
?
137
0
0
0
0
0
0
0
0
0
0
0
0
0
9.11974006×10¹⁰
0
0
0
0
70080744
0
0
0
0
≥1
0
0
0
?
138
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1.08876553×10¹¹
0
0
0
0
3.33174285×10¹³
0
0
0
0
≥1
0
0
0
?
139
0
0
0
0
0
0
0
0
0
0
0
0
0
1.29982811×10¹¹
0
0
0
0
86933996
0
0
0
0
≥1
0
0
0
?
140
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
1.55182193×10¹¹
0
18816
8.14328246×10¹²
59149756
141125236
2.56496684×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
141
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1.85264663×10¹¹
0
0
0
0
6.41356997×10¹³
0
0
0
0
≥1
0
0
0
?
142
0
0
0
0
0
0
0
0
0
0
0
0
0
2.21180212×10¹¹
0
0
0
0
174274846
0
0
0
0
≥1
0
0
0
?
143
0
0
0
0
0
0
0
0
0
0
0
0
0
2.64058746×10¹¹
0
0
0
0
283510154
0
0
0
0
≥1
0
0
0
?
144
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
3.15247475×10¹¹
0
0
0
0
1.23460754×10¹⁴
0
0
0
0
≥1
0
0
0
?
145
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
3.76362405×10¹¹
0
796232
2.26810637×10¹³
2.48846988×10¹⁰
349904428
8.07784249×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
146
0
0
0
0
0
0
0
0
0
0
0
0
0
4.49323410×10¹¹
0
0
0
0
567901778
0
0
0
0
≥1
0
0
0
?
147
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
5.36427687×10¹¹
0
0
0
0
2.37661317×10¹⁴
0
0
0
0
≥1
0
0
0
?
148
0
0
0
0
0
0
0
0
0
0
0
0
0
6.40421151×10¹¹
0
0
0
0
700415476
0
0
0
0
≥1
0
0
0
?
149
0
0
0
0
0
0
0
0
0
0
0
0
0
7.64570885×10¹¹
0
0
0
0
1.13592184×10¹⁰
0
0
0
0
≥1
0
0
0
?
150
0
1
0
≥1
≥1
0
1024
≥1
≥1
59049
1
59049
1048576
9.12790092×10¹¹
60466176
10465989
6.31723960×10¹³
1.01145196×10¹¹
4.57496789×10¹⁴
2.54413782×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
151
0
0
0
0
0
0
0
0
0
0
0
0
0
1.08974456×10¹²
0
0
0
0
1.40236315×10¹⁰
0
0
0
0
≥1
0
0
0
?
152
0
0
0
0
0
0
0
0
0
0
0
0
0
1.30099857×10¹²
0
0
0
0
2.26880093×10¹⁰
0
0
0
0
≥1
0
0
0
?
153
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1.55321124×10¹²
0
0
0
0
8.80678351×10¹⁴
0
0
0
0
≥1
0
0
0
?
154
0
0
0
0
0
0
0
0
0
0
0
0
0
1.85431544×10¹²
0
0
0
0
2.80509964×10¹⁰
0
0
0
0
≥1
0
0
0
?
155
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
2.21378866×10¹²
0
277578
1.75950813×10¹⁴
939499380
4.52845910×10¹⁰
8.01336576×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
156
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
2.64295580×10¹²
0
0
0
0
1.69529927×10¹⁵
0
0
0
0
≥1
0
0
0
?
157
0
0
0
0
0
0
0
0
0
0
0
0
0
3.15531401×10¹²
0
0
0
0
5.59640322×10¹⁰
0
0
0
0
≥1
0
0
0
?
158
0
0
0
0
0
0
0
0
0
0
0
0
0
3.76699990×10¹²
0
0
0
0
9.02693837×10¹⁰
0
0
0
0
≥1
0
0
0
?
159
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
4.49727125×10¹²
0
0
0
0
3.26343836×10¹⁵
0
0
0
0
≥1
0
0
0
?
160
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
5.36910268×10¹²
0
6133034
4.90066713×10¹⁴
3.02841504×10¹¹
1.11601810×10¹¹
2.52415585×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
161
0
0
0
0
0
0
0
0
0
0
0
0
0
6.40995571×10¹²
0
0
0
0
1.79754615×10¹¹
0
0
0
0
≥1
0
0
0
?
162
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
7.65258527×10¹²
0
0
0
0
6.28209592×10¹⁵
0
0
0
0
≥1
0
0
0
?
163
0
0
0
0
0
0
0
0
0
0
0
0
0
9.13610259×10¹²
0
0
0
0
2.22127282×10¹¹
0
0
0
0
≥1
0
0
0
?
164
0
0
0
0
0
0
0
0
0
0
0
0
0
1.09072269×10¹³
0
0
0
0
3.57477799×10¹¹
0
0
0
0
≥1
0
0
0
?
165
0
1
0
≥1
≥1
0
2048
≥1
≥1
177147
1
177147
4194304
1.30216879×10¹³
362797056
53298597
1.36495750×10¹⁵
1.01939427×10¹²
1.20929930×10¹⁶
7.95137374×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
166
0
0
0
0
0
0
0
0
0
0
0
0
0
1.55460583×10¹³
0
0
0
0
4.41787959×10¹¹
0
0
0
0
≥1
0
0
0
?
167
0
0
0
0
0
0
0
0
0
0
0
0
0
1.85598122×10¹³
0
0
0
0
7.10154695×10¹¹
0
0
0
0
≥1
0
0
0
?
168
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
2.21577905×10¹³
0
0
0
0
2.32789324×10¹⁶
0
0
0
0
≥1
0
0
0
?
169
0
0
0
0
0
0
0
0
0
0
0
0
0
2.64532852×10¹³
0
0
0
0
8.78154732×10¹¹
0
0
0
0
≥1
0
0
0
?
170
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
3.15815002×10¹³
0
3265658
3.80174562×10¹⁵
1.43882323×10¹¹
1.40976762×10¹²
2.50490390×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
171
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
3.77038488×10¹³
0
0
0
0
4.48117885×10¹⁶
0
0
0
0
≥1
0
0
0
?
172
0
0
0
0
0
0
0
0
0
0
0
0
0
4.50130975×10¹³
0
0
0
0
1.74325653×10¹²
0
0
0
0
≥1
0
0
0
?
173
0
0
0
0
0
0
0
0
0
0
0
0
0
5.37392907×10¹³
0
0
0
0
2.79599335×10¹²
0
0
0
0
≥1
0
0
0
?
174
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
6.41571341×10¹³
0
0
0
0
8.62623843×10¹⁶
0
0
0
0
≥1
0
0
0
?
175
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
7.65945977×10¹³
0
44207830
1.05888057×10¹⁶
3.60064025×10¹²
3.45837088×10¹²
7.89152883×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
176
0
0
0
0
0
0
0
0
0
0
0
0
0
9.14431396×10¹³
0
0
0
0
5.54135700×10¹²
0
0
0
0
≥1
0
0
0
?
177
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1.09170231×10¹⁴
0
0
0
0
1.66054493×10¹⁷
0
0
0
0
≥1
0
0
0
?
178
0
0
0
0
0
0
0
0
0
0
0
0
0
1.30333888×10¹⁴
0
0
0
0
6.85348977×10¹²
0
0
0
0
≥1
0
0
0
?
179
0
0
0
0
0
0
0
0
0
0
0
0
0
1.55600273×10¹⁴
0
0
0
0
1.09719409×10¹³
0
0
0
0
≥1
0
0
0
?
180
0
1
0
≥1
≥1
0
4096
≥1
≥1
531441
1
531441
16777216
1.85764829×10¹⁴
2.17678233×10¹⁰
272221943
2.94924536×10¹⁶
1.03033425×10¹³
3.19653759×10¹⁷
2.48628436×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
181
0
0
0
0
0
0
0
0
0
0
0
0
0
2.21777028×10¹⁴
0
0
0
0
1.35709855×10¹³
0
0
0
0
≥1
0
0
0
?
182
0
0
0
0
0
0
0
0
0
0
0
0
0
2.64770505×10¹⁴
0
0
0
0
2.17071910×10¹³
0
0
0
0
≥1
0
0
0
?
183
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
3.16098717×10¹⁴
0
0
0
0
6.15331314×10¹⁷
0
0
0
0
≥1
0
0
0
?
184
0
0
0
0
0
0
0
0
0
0
0
0
0
3.77377301×10¹⁴
0
0
0
0
2.68580634×10¹³
0
0
0
0
≥1
0
0
0
?
185
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
4.50535334×10¹⁴
0
32983408
8.21438076×10¹⁶
2.12504483×10¹²
4.29209560×10¹³
7.83354837×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
186
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
5.37875745×10¹⁴
0
0
0
0
1.18450861×10¹⁸
0
0
0
0
≥1
0
0
0
?
187
0
0
0
0
0
0
0
0
0
0
0
0
0
6.42147807×10¹⁴
0
0
0
0
5.31082489×10¹³
0
0
0
0
≥1
0
0
0
?
188
0
0
0
0
0
0
0
0
0
0
0
0
0
7.66634052×10¹⁴
0
0
0
0
8.48073043×10¹³
0
0
0
0
≥1
0
0
0
?
189
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
9.15253047×10¹⁴
0
0
0
0
2.28017103×10¹⁸
0
0
0
0
≥1
0
0
0
?
190
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
1.09268314×10¹⁵
0
303424822
2.28790904×10¹⁷
4.20397030×10¹³
1.04949941×10¹⁴
2.46821484×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
191
0
0
0
0
0
0
0
0
0
0
0
0
0
1.30450979×10¹⁵
0
0
0
0
1.67474339×10¹⁴
0
0
0
0
≥1
0
0
0
?
192
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
1.55740085×10¹⁵
0
0
0
0
4.38931374×10¹⁸
0
0
0
0
≥1
0
0
0
?
193
0
0
0
0
0
0
0
0
0
0
0
0
0
1.85931719×10¹⁵
0
0
0
0
2.07256612×10¹⁴
0
0
0
0
≥1
0
0
0
?
194
0
0
0
0
0
0
0
0
0
0
0
0
0
2.21976284×10¹⁵
0
0
0
0
3.30491049×10¹⁴
0
0
0
0
≥1
0
0
0
?
195
0
1
0
≥1
≥1
0
8192
≥1
≥1
1594323
1
1594323
67108864
2.65008399×10¹⁵
1.30606940×10¹¹
1.39643407×10¹⁰
6.37239489×10¹⁷
1.04470544×10¹⁴
8.44939904×10¹⁸
7.77719185×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
196
0
0
0
0
0
0
0
0
0
0
0
0
0
3.16382699×10¹⁵
0
0
0
0
4.09034279×10¹⁴
0
0
0
0
≥1
0
0
0
?
197
0
0
0
0
0
0
0
0
0
0
0
0
0
3.77716370×10¹⁵
0
0
0
0
6.51781479×10¹⁴
0
0
0
0
≥1
0
0
0
?
198
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
4.50940119×10¹⁵
0
0
0
0
1.62650355×10¹⁹
0
0
0
0
≥1
0
0
0
?
199
0
0
0
0
0
0
0
0
0
0
0
0
0
5.38358983×10¹⁵
0
0
0
0
8.06860574×10¹⁴
0
0
0
0
≥1
0
0
0
?
200
0
1
0
0
≥1
0
0
≥1
0
0
1
0
0
6.42724769×10¹⁵
0
298685688
1.77487023×10¹⁸
3.03410530×10¹³
1.28480420×10¹⁵
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
5k
x
3k
5k
x
15k
5k
3k
15k
5k
15k
15k
all
15k
5k
5k
5k
all
5k
?
?
?
?
?
?
?

Formulas

$N(w; h)$ - number of ways to tile $w\times h$ rectangle (including symmetric solutions)

$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise

$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)

$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$

$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$

$G(N(3); x) = \frac{x^5}{1 - x^5} \tag{1}$

$G(N(5); x) = \frac{x^3}{1 - x^3} \tag{2}$

$G(N(6); x) = \frac{x^5}{1 - x^5} \tag{3}$

$G(N(8); x) = \frac{2x^{15}}{1 - 2x^{15}} \tag{4}$

$G(N(9); x) = \frac{x^5}{1 - x^5} \tag{5}$

$G(N(10); x) = \frac{x^3}{1 - x^3} \tag{6}$

$G(N(11); x) = \frac{3x^{15}}{1 - 3x^{15}} \tag{7}$

$G(N(12); x) = \frac{x^5}{1 - x^5} \tag{8}$

$G(N(13); x) = \frac{3x^{15}}{1 - 3x^{15}} \tag{9}$

$G(N(14); x) = \frac{4x^{15}}{1 - 4x^{15}} \tag{10}$

$G(N(15); x) = \frac{x^3 + x^5}{1 - x^3 - x^5} \tag{11}$

$G(N(16); x) = \frac{6x^{15}}{1 - 6x^{15}} \tag{12}$

$G(N(18); x) = \frac{x^5 - 4x^{10} + 10x^{15} - 12x^{20} + 13x^{25} - 8x^{30} + 2x^{35}}{1 - 5x^5 + 10x^{10} - 14x^{15} + 13x^{20} - 13x^{25} + 8x^{30} - 2x^{35}} \tag{13}$

Attributions

  1. Prime rectangle list taken from http://www.recmath.com/PolyPages/PolyPages/index.htm?5pairs.htm

See Also

U pentomino and V pentominoU pentomino and Y pentomino