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.


X pentomino and F hexomino

Prime rectangles: ≥ 120.

Smallest rectangle tilings

Smallest rectangle and smallest square (7x7):

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
1-6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N>0
1-6
0
7
0
2
8
0
0
0
9
0
0
0
0
10
0
0
0
0
0
11
0
64
0
0
8
0
12
0
0
0
0
0
0
0
13
0
0
0
0
0
32
0
0
14
0
4
0
0
0
8192
0
0
≥1
15
0
0
0
0
1664
0
0
320
≥1
0
16
0
0
0
0
0
4096
0
796
≥1
≥1
≥1
17
0
0
0
0
0
2320
0
0
≥1
≥1
≥1
≥1
18
0
256
0
0
0
36352
0
128
≥1
≥1
≥1
≥1
≥1
19
0
1152
0
0
57344
933888
0
137728
≥1
≥1
≥1
≥1
≥1
≥1
20
0
0
0
0
8192
25280
224
2856832
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
8
0
0
3200
807776
23040
11136
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
22
0
4096
0
0
64
215968
1536
19328
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
23
0
36864
0
0
79104
852480
6656
27464496
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
24
0
0
0
0
1280
432640
389376
1561280
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
25
0
768
0
0
0
9568768
956416
31430944
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
26
0
3456
0
0
34816
174980528
258048
6555504
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
27
0
0
0
0
16963584
9083392
5137408
1.08898518×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
28
0
1296
0
0
1036288
200107954
23679760
975921592
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
29
0
36096
0
0
2048
102999296
30849792
5.84324580×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
30
0
247104
0
0
4909120
1.06757049×10¹⁰
68503552
1.43226459×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
0
497664
0
0
597868544
1.03390569×10¹¹
407418624
2.42843283×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
32
0
2048
0
0
107472128
2.60113184×10¹⁰
958349680
3.80898458×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
33
0
271360
0
0
24218144
2.59720092×10¹¹
1.54361961×10¹⁰
3.40739484×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
34
0
3538944
0
0
252538880
7.27632405×10¹⁰
6.16492662×10¹⁰
3.23555609×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
35
0
15930400
0
0
656297728
4.12249581×10¹¹
2.02110342×10¹¹
3.99667276×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
36
0
144384
0
0
197200896
2.98679888×10¹¹
3.94150570×10¹¹
8.08240055×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
37
0
988416
0
0
197988864
3.11917553×10¹²
1.01594930×10¹²
6.86872515×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
38
0
1990656
0
0
4.53015347×10¹⁰
2.58171633×10¹³
3.55004925×10¹²
3.27887665×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
39
0
168960
0
0
1.27719742×10¹²
6.32517165×10¹²
8.74261380×10¹²
1.84995495×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
40
0
4332032
0
0
2.08982146×10¹¹
5.65848041×10¹³
1.94345372×10¹³
1.83242226×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
41
0
38264832
0
0
2.00458368×10¹⁰
3.06705341×10¹³
5.91304599×10¹³
6.65956554×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
42
0
142347328
0
0
8.68406881×10¹¹
2.20831319×10¹⁴
1.68061814×10¹⁴
4.28640712×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
43
0
191568896
0
0
4.43896806×10¹³
1.07564962×10¹⁵
3.89454137×10¹⁴
3.27960563×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
44
0
20037376
0
0
9.90678499×10¹²
8.83490393×10¹⁴
1.01809177×10¹⁵
3.57511931×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
45
0
308651008
0
0
1.94754003×10¹²
5.14606073×10¹⁵
2.96177442×10¹⁵
7.49732846×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
46
0
2.03977267×10¹⁰
0
0
4.31394154×10¹³
2.33250011×10¹⁵
7.49316759×10¹⁵
4.46255645×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
47
0
6.14066252×10¹⁰
1440
0
5.27176839×10¹³
1.22512382×10¹⁶
1.83725602×10¹⁶
4.31629863×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
48
0
203610240
0
0
3.17125370×10¹³
1.01426387×10¹⁶
5.07121164×10¹⁶
1.57490102×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
49
0
726630528
0
0
3.37800336×10¹³
7.02277663×10¹⁶
1.35873933×10¹⁷
8.57536725×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
50
0
972595200
29952
0
7.51435882×10¹⁴
3.33450736×10¹⁷
3.35956657×10¹⁷
7.47973522×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
51
0
460564480
0
0
8.41362506×10¹⁵
2.22442317×10¹⁷
8.73883129×10¹⁷
2.06846504×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
52
0
4.95093760×10¹⁰
0
0
2.33639283×10¹⁵
1.20891484×10¹⁸
2.36316258×10¹⁸
2.27538387×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
53
0
2.61227499×10¹¹
290304
0
6.85624993×10¹⁴
8.17774554×10¹⁷
6.03186176×10¹⁸
5.90916776×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
54
0
6.77040547×10¹¹
0
0
9.62149278×10¹⁵
4.41576970×10¹⁸
1.52475875×10¹⁹
6.74549499×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
55
0
7.06852541×10¹¹
0
0
2.83620007×10¹⁷
1.20153281×10¹⁹
4.03660994×10¹⁹
3.70763259×10²¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
56
0
2.70705666×10¹¹
2138112
0
7.70586349×10¹⁶
2.11689859×10¹⁹
1.05358729×10²⁰
2.91732476×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
57
0
2.21472755×10¹²
0
0
2.08480052×10¹⁶
8.34491398×10¹⁹
2.66292143×10²⁰
1.02131373×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
58
0
9.82342837×10¹²
0
0
4.64578696×10¹⁷
6.63335173×10¹⁹
6.87658869×10²⁰
4.60961239×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
59
0
2.23676155×10¹³
13639680
0
4.22521798×10¹⁷
2.83661763×10²⁰
1.80079015×10²¹
4.05387626×10²³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
60
0
1.70203478×10¹²
28800
0
3.35436762×10¹⁷
2.70435712×10²⁰
4.60172125×10²¹
2.13902940×10²³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
61
0
4.20200399×10¹²
0
0
4.67491688×10¹⁷
1.40179484×10²¹
1.17231379×10²²
8.69539739×10²³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
62
0
4.58672895×10¹²
79626240
0
8.04122335×10¹⁸
4.26120261×10²¹
3.04180963×10²²
1.15927826×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
63
0
5.71852677×10¹²
783360
0
5.15495322×10¹⁹
5.72783728×10²¹
7.83706175×10²²
1.91219174×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
64
0
3.84686063×10¹³
0
0
2.01734976×10¹⁹
2.19478824×10²²
1.99272336×10²³
2.33747562×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
65
0
1.45131127×10¹⁴
437649408
0
1.17319855×10¹⁹
2.05530982×10²²
5.11300295×10²³
4.87051791×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
66
0
2.91355967×10¹⁴
10450944
0
8.61393674×10¹⁹
8.68977150×10²²
1.31834736×10²⁴
8.16783093×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
67
0
2.64445511×10¹⁴
0
0
1.66744982×10²¹
1.61076979×10²³
3.36311925×10²⁴
3.76518143×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
68
0
2.28724413×10¹⁴
2.30267289×10¹⁰
0
5.40892551×10²⁰
4.35215100×10²³
8.57038201×10²⁴
2.22952657×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
69
0
1.27695957×10¹⁵
99385344
0
2.38834488×10²⁰
1.25863172×10²⁴
2.19996760×10²⁵
1.14931648×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
70
0
4.28042073×10¹⁵
7200
0
4.01593127×10²¹
1.63812450×10²⁴
5.62682453×10²⁵
4.38464131×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
71
0
7.92673175×10¹⁵
1.17209825×10¹¹
0
3.32688675×10²¹
5.64157340×10²⁴
1.43112863×10²⁶
3.50774156×10²⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
72
0
1.11056070×10¹⁵
780042240
0
2.95663515×10²¹
6.49929562×10²⁴
3.65365572×10²⁶
2.41713309×10²⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
73
0
2.14242995×10¹⁵
817920
0
5.36316904×10²¹
2.67611250×10²⁵
9.34222191×10²⁶
8.04093842×10²⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
74
0
2.28281576×10¹⁵
5.81330534×10¹¹
0
6.93305702×10²²
5.91371822×10²⁵
2.37726778×10²⁷
1.45511620×10²⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
75
0
4.96023449×10¹⁵
5.41458432×10¹⁰
0
3.02775786×10²³
1.24811760×10²⁶
6.04660658×10²⁷
1.59928057×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
76
0
2.42499214×10¹⁶
23551488
0
1.51385022×10²³
3.67125031×10²⁶
1.54228508×10²⁸
2.16265928×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
77
0
7.14294246×10¹⁶
2.82360545×10¹²
0
1.43408846×10²³
4.81864229×10²⁶
3.92632319×10²⁸
3.95544079×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
78
0
1.18634835×10¹⁷
3.45165004×10¹¹
0
6.84680289×10²³
1.65251812×10²⁷
9.97068342×10²⁸
8.64940758×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
79
0
1.04752173×10¹⁷
374611968
0
9.32415662×10²⁴
2.65466829×10²⁷
2.53576086×10²⁹
3.56413107×10³¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
80
0
1.52272018×10¹⁷
1.34808286×10¹³
0
3.56383971×10²⁴
8.33343285×10²⁷
6.45123323×10²⁹
1.65269083×10³²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
81
0
6.47423640×10¹⁷
2.06712078×10¹²
0
2.49619786×10²⁴
1.91962538×10²⁸
1.63753238×10³⁰
1.17033040×10³²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
82
0
1.75556568×10¹⁸
4.26155212×10¹⁰
0
3.04853308×10²⁵
3.60484841×10²⁸
4.15544207×10³⁰
4.05573362×10³²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
83
0
2.78390394×10¹⁸
6.34420385×10¹³
0
2.52837454×10²⁵
1.02736995×10²⁹
1.05554817×10³¹
2.88062847×10³³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
84
0
6.26440003×10¹⁷
1.18025197×10¹³
0
2.38215459×10²⁵
1.46581343×10²⁹
2.67840981×10³¹
2.45184216×10³³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
85
0
1.02472340×10¹⁸
3.93265152×10¹¹
0
5.28371337×10²⁵
4.98815856×10²⁹
6.78826679×10³¹
7.10564690×10³³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
86
0
1.22959485×10¹⁸
2.94935634×10¹⁴
0
5.25324422×10²⁶
9.35055621×10²⁹
1.72132243×10³²
1.59945343×10³⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
87
0
3.48456696×10¹⁸
6.48976269×10¹³
0
1.73458519×10²⁷
2.47916152×10³⁰
4.36454931×10³²
1.26609677×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
88
0
1.33915122×10¹⁹
3.15125268×10¹²
0
1.04756353×10²⁷
6.01549479×10³⁰
1.10540659×10³³
1.87881858×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
89
0
3.25124486×10¹⁹
1.35678715×10¹⁵
0
1.43336353×10²⁷
1.05044776×10³¹
2.79902987×10³³
3.23811295×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
90
0
4.70573682×10¹⁹
3.46140811×10¹⁴
0
5.05765844×10²⁷
3.03137060×10³¹
7.08943835×10³³
8.45141829×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
91
0
4.46291994×10¹⁹
2.28226970×10¹³
0
5.05698631×10²⁸
4.99750579×10³¹
1.79449885×10³⁴
3.20838722×10³⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
92
0
8.75868100×10¹⁹
6.18487941×10¹⁵
0
2.26622209×10²⁸
1.54321418×10³²
4.53943269×10³⁴
1.21239880×10³⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
93
0
3.02112572×10²⁰
1.80034298×10¹⁵
0
2.32715499×10²⁸
3.11327123×10³²
1.14842995×10³⁵
1.12175442×10³⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
94
0
6.93951914×10²⁰
1.53231388×10¹⁴
0
2.12623541×10²⁹
7.31306767×10³²
2.90489407×10³⁵
3.69293818×10³⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
95
0
9.81531922×10²⁰
2.79691131×10¹⁶
0
1.84154966×10²⁹
1.79810525×10³³
7.34322645×10³⁵
2.28700149×10³⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
96
0
3.22094608×10²⁰
9.16831772×10¹⁵
0
1.83420978×10²⁹
3.13554626×10³³
1.85591021×10³⁶
2.32086767×10³⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
97
0
4.79089550×10²⁰
9.70057227×10¹⁴
0
4.64218130×10²⁹
9.13175228×10³³
4.69055732×10³⁶
6.13574200×10³⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
98
0
6.92570006×10²⁰
1.25592796×10¹⁷
0
3.65845132×10³⁰
1.65703437×10³⁴
1.18499951×10³⁷
1.60656997×10³⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
99
0
2.12855581×10²¹
4.58584175×10¹⁶
0
9.79853180×10³⁰
4.68243410×10³⁴
2.99260337×10³⁷
9.73874281×10³⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
100
0
6.75359354×10²¹
5.85975017×10¹⁵
0
6.92673444×10³⁰
1.00728698×10³⁵
7.55709074×10³⁷
1.56986615×10⁴⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
101
0
1.40518869×10²²
5.60448867×10¹⁷
0
1.25316772×10³¹
2.15080931×10³⁵
1.90798958×10³⁸
2.68882227×10⁴⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
102
0
1.85790334×10²²
2.25852045×10¹⁷
0
3.56012242×10³¹
5.43296408×10³⁵
4.81543190×10³⁸
7.82661079×10⁴⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
103
0
2.01408345×10²²
3.40677662×10¹⁶
0
2.69511461×10³²
9.87940881×10³⁵
1.21511289×10³⁹
2.78185214×10⁴¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
104
0
4.56737936×10²²
2.48707356×10¹⁸
0
1.41943139×10³²
2.81231248×10³⁶
3.06586372×10³⁹
8.91275286×10⁴¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
105
0
1.33334618×10²³
1.09742762×10¹⁸
0
1.96851520×10³²
5.42776275×10³⁶
7.73338223×10³⁹
1.03291412×10⁴²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
106
0
2.68602032×10²³
1.91866190×10¹⁷
0
1.39918341×10³³
1.40756192×10³⁷
1.95018855×10⁴⁰
3.31154385×10⁴²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
107
0
3.50533782×10²³
1.09819209×10¹⁹
0
1.28719567×10³³
3.13356156×10³⁷
4.91738232×10⁴⁰
1.77784001×10⁴³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
108
0
1.56064602×10²³
5.26972401×10¹⁸
0
1.37707327×10³³
6.40165771×10³⁷
1.23969259×10⁴¹
2.09665212×10⁴³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
109
0
2.24779139×10²³
1.05195701×10¹⁸
0
3.74529619×10³³
1.65190336×10³⁸
3.12453649×10⁴¹
5.23686805×10⁴³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
110
0
3.89660833×10²³
4.82756238×10¹⁹
0
2.40511802×10³⁴
3.13047474×10³⁸
7.87386415×10⁴¹
1.51382259×10⁴⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
111
0
1.17731010×10²⁴
2.50409964×10¹⁹
0
5.49844521×10³⁴
8.61880230×10³⁸
1.98396428×10⁴²
7.39713037×10⁴⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
112
0
3.19194836×10²⁴
5.63677890×10¹⁸
0
4.47791482×10³⁴
1.75693278×10³⁹
4.99794438×10⁴²
1.28100645×10⁴⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
113
0
5.87374050×10²⁴
2.11365123×10²⁰
0
9.96668671×10³⁴
4.20038830×10³⁹
1.25883179×10⁴³
2.26009510×10⁴⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
114
0
7.41427571×10²⁴
1.17886390×10²⁰
0
2.42286060×10³⁵
9.67507502×10³⁹
3.17021734×10⁴³
6.97769335×10⁴⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
115
0
9.35514924×10²⁴
2.96104632×10¹⁹
0
1.42524960×10³⁶
1.95594889×10⁴⁰
7.98257184×10⁴³
2.34494660×10⁴⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
116
0
2.22303921×10²⁵
9.22079229×10²⁰
0
8.87947628×10³⁵
5.08955838×10⁴⁰
2.00963488×10⁴⁴
6.61635864×10⁴⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
117
0
5.66690136×10²⁵
5.50358033×10²⁰
0
1.54250573×10³⁶
9.98216655×10⁴⁰
5.05860133×10⁴⁴
9.24668749×10⁴⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
118
0
1.02952890×10²⁶
1.52873837×10²⁰
0
8.83759553×10³⁶
2.62614451×10⁴¹
1.27317478×10⁴⁵
2.92093359×10⁴⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
119
0
1.27521772×10²⁶
4.00947571×10²¹
0
8.68212669×10³⁶
5.55091395×10⁴¹
3.20388160×10⁴⁵
1.36579276×10⁴⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
120
0
7.29463895×10²⁵
2.55010046×10²¹
0
1.01491230×10³⁷
1.25765047×10⁴²
8.06121126×10⁴⁵
1.83359747×10⁴⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
121
0
1.07067052×10²⁶
7.77307501×10²⁰
0
2.83438470×10³⁷
2.97680067×10⁴²
2.02801624×10⁴⁶
4.44301939×10⁴⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
122
0
2.12943860×10²⁶
1.73832325×10²²
0
1.51883609×10³⁸
6.04195089×10⁴²
5.10134129×10⁴⁶
1.36182474×10⁴⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
123
0
6.04973142×10²⁶
1.17357610×10²²
0
3.08222902×10³⁸
1.56862014×10⁴³
1.28302447×10⁴⁷
5.60438365×10⁴⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
124
0
1.43907012×10²⁷
3.89916212×10²¹
0
2.87092981×10³⁸
3.18504391×10⁴³
3.22650244×10⁴⁷
1.03131558×10⁵⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
125
0
2.40767453×10²⁷
7.51662361×10²²
0
7.39375644×10³⁸
7.94448317×10⁴³
8.11293466×10⁴⁷
1.91328412×10⁵⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
126
0
3.01662142×10²⁷
5.36758051×10²²
0
1.60806176×10³⁹
1.73479454×10⁴⁴
2.03970570×10⁴⁸
6.05058659×10⁵⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
127
0
4.36530974×10²⁷
1.93239851×10²²
0
7.53810245×10³⁹
3.81017770×10⁴⁴
5.12747394×10⁴⁸
1.93570394×10⁵¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
128
0
1.02964672×10²⁸
3.24248888×10²³
0
5.59043010×10³⁹
9.20338709×10⁴⁴
1.28881743×10⁴⁹
4.97993496×10⁵¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
129
0
2.34917908×10²⁸
2.44116529×10²³
0
1.13915923×10⁴⁰
1.88589392×10⁴⁵
3.23913604×10⁴⁹
8.10858269×10⁵¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
130
0
3.93917779×10²⁸
9.47332730×10²²
0
5.42188137×10⁴⁰
4.82156822×10⁴⁵
8.13983584×10⁴⁹
2.53379658×10⁵²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
131
0
4.74000250×10²⁸
1.39573082×10²⁴
0
5.69054707×10⁴⁰
1.00565652×10⁴⁶
2.04529158×10⁵⁰
1.04401181×10⁵³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
132
0
3.34557972×10²⁸
1.10452918×10²⁴
0
7.34773815×10⁴⁰
2.40038603×10⁴⁶
5.13865247×10⁵⁰
1.56734412×10⁵³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
133
0
5.17219134×10²⁸
4.59881400×10²³
0
2.04314537×10⁴¹
5.38395559×10⁴⁶
1.29090943×10⁵¹
3.75582688×10⁵³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
134
0
1.11854801×10²⁹
5.99635803×10²⁴
0
9.32548517×10⁴¹
1.16412962×10⁴⁷
3.24262446×10⁵¹
1.18421886×10⁵⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
135
0
2.93967725×10²⁹
4.97399310×10²⁴
0
1.73390877×10⁴²
2.84731294×10⁴⁷
8.14431697×10⁵¹
4.26062651×10⁵⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
136
0
6.27003549×10²⁹
2.21269340×10²⁴
0
1.83954122×10⁴²
5.92194431×10⁴⁷
2.04535725×10⁵²
8.24884200×10⁵⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
137
0
9.78169133×10²⁹
2.57171937×10²⁵
0
5.20470072×10⁴²
1.47400731×10⁴⁸
5.13617891×10⁵²
1.62292159×10⁵⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
138
0
1.25385271×10³⁰
2.23022049×10²⁵
0
1.04652741×10⁴³
3.15160872×10⁴⁸
1.28964430×10⁵³
5.14001678×10⁵⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
139
0
2.01789327×10³⁰
1.05601961×10²⁵
0
4.01434045×10⁴³
7.28209265×10⁴⁸
3.23787013×10⁵³
1.57395738×10⁵⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
140
0
4.60099228×10³⁰
1.10126081×10²⁶
0
3.55041035×10⁴³
1.67045327×10⁴⁹
8.12845700×10⁵³
3.80522062×10⁵⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
141
0
9.58737991×10³⁰
9.95989358×10²⁵
0
8.03537660×10⁴³
3.58858764×10⁴⁹
2.04040898×10⁵⁴
7.00041270×10⁵⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
142
0
1.51297203×10³¹
5.00263284×10²⁵
0
3.25904603×10⁴⁴
8.79842368×10⁴⁹
5.12139529×10⁵⁴
2.16384093×10⁵⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
143
0
1.80233740×10³¹
4.70935646×10²⁶
0
3.64974086×10⁴⁴
1.85609117×10⁵⁰
1.28534980×10⁵⁵
7.97977508×10⁵⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
144
0
1.52331703×10³¹
4.43160320×10²⁶
0
5.22024100×10⁴⁴
4.49318085×10⁵⁰
3.22564590×10⁵⁵
1.31841406×10⁵⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
145
0
2.51366515×10³¹
2.35378049×10²⁶
0
1.41860502×10⁴⁵
9.81665991×10⁵⁰
8.09423362×10⁵⁵
3.16509737×10⁵⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
146
0
5.64326318×10³¹
2.01143577×10²⁷
0
5.61610701×10⁴⁵
2.21965558×10⁵¹
2.03094942×10⁵⁶
1.00452955×10⁵⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
147
0
1.36832484×10³²
1.96511804×10²⁷
0
9.82618327×10⁴⁵
5.18083295×10⁵¹
5.09550114×10⁵⁶
3.26004019×10⁵⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
148
0
2.66681409×10³²
1.10054690×10²⁷
0
1.18134204×10⁴⁶
1.11368846×10⁵²
1.27832116×10⁵⁷
6.58478929×10⁵⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
149
0
3.97024627×10³²
8.58197465×10²⁷
0
3.51957044×10⁴⁶
2.70894537×10⁵²
3.20670701×10⁵⁷
1.37405111×10⁶⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
150
0
5.30810471×10³²
8.68659041×10²⁷
0
6.70338614×10⁴⁶
5.79983692×10⁵²
8.04350007×10⁵⁷
4.30032988×10⁶⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
all
all
x
all
all
all
all
?
?
?
?
?
?
?

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 $A(w; h)$

$G(N(7); x) = \frac{2x^7 + 64x^{11} - 40x^{24} - 180x^{25} + 5760x^{29} + 25920x^{30}}{1 - 2x^7 - 64x^{11} - 576x^{12} - 20x^{17} - 90x^{18} + 576x^{19} + 18432x^{23} + 82984x^{24} + 180x^{25} + 5760x^{29} + 25920x^{30} - 11520x^{36} - 51840x^{37}} \tag{1}$

$G(N(8); x) = \frac{1440x^{47} + 12672x^{50} - 5760x^{63} - 36000x^{71} - 316800x^{74} - 115200x^{77} - 5760x^{79} - 1013760x^{80} - 50688x^{82} + 144000x^{87} + 92160x^{101} + 144000x^{103} + 811008x^{104} + 1267200x^{106}}{1 - 12x^3 + 48x^6 - 64x^9 - 20x^{13} + 108x^{16} + 96x^{19} - 832x^{22} - 5x^{23} - 25x^{24} - 4x^{26} + 300x^{27} + 32x^{29} - 1280x^{30} + 1212x^{32} + 2560x^{33} - 3792x^{35} - 3840x^{36} + 500x^{37} - 192x^{38} + 5140x^{39} - 2700x^{40} + 256x^{41} - 160x^{42} - 900x^{43} + 400x^{45} + 10480x^{46} - 1315x^{47} - 448x^{48} + 10560x^{49} - 6812x^{50} - 256x^{51} + 26880x^{52} + 50288x^{53} + 3136x^{54} + 20x^{55} - 29980x^{56} - 768x^{57} + 16x^{58} + 105440x^{59} + 3072x^{60} + 192x^{61} - 198080x^{62} + 1164x^{63} - 7424x^{64} + 512000x^{65} - 19040x^{66} + 19280x^{67} - 10000x^{69} + 8256x^{70} + 36000x^{71} + 11200x^{72} - 8448x^{73} + 172800x^{74} + 6800x^{75} - 21504x^{76} - 1152320x^{77} - 76480x^{78} + 5260x^{79} + 552704x^{80} - 21760x^{81} + 27248x^{82} - 4063552x^{83} + 56320x^{84} - 207552x^{85} + 162304x^{86} - 144000x^{87} + 185600x^{88} - 414720x^{89} + 576000x^{90} - 512000x^{91} - 320x^{99} - 92160x^{101} - 256x^{102} - 144000x^{103} - 442368x^{104} + 17408x^{105} - 691200x^{106} + 3244032x^{107} - 45056x^{108} + 5068800x^{109}} \tag{2}$

See Also

X pentomino and D hexominoX pentomino and L hexomino