The best known packings of equal circles in a rectangle with variable aspect ratio (complete up to N = 500)

Last update: 10-Dec-2011


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Download

You may download ASCII files which contain all the values of radius, ratio etc. by using the links given in the table header below.
All coordinates of all packings are packed as ASCII files here.
All packings are stored as nice PDF files here.
All contact graphs of all packings are stored as nice PDF files here.
  For industrial applications, for instance if a machine has to do an important job at every circle center,
it is useful to know a tour visiting each of the circle centers once which is of minimal length.
This problem is known as the "Traveling Salesman Problem" (TSP). Thus (very near) optimal tours are provided for every packing.
All optimal TSP tours of all packings are stored as nice PDF files here.


Results

The table below summarizes the current status of the search.
Please use the links in the following table to view a picture for a certain configuration.
Furthermore, note that for certain values of N several distinct optimal configurations exist; however, only one is shown here.
Proven optimal packings are indicated by a radius in bold face type.

Legend:
Please note that all packings (including their coordinates, of course) are normalized such that their width (i.e. the horizontal dimension) is equal 1.
N
the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
class
please refer to [4]
radius
of the circles in the rectangle
height
= height of the rectangle (W = 1)
density
ratio of total area occupied by the circles to container area (for an infinite hexagonal packing you get the well-known value ρ = Pi/(2*sqrt(3))=0.90689968211)
contacts
number of contacts between circles and container and between the circles themselves, respectively
loose
number of circles that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of circles that have contact to the container (rattlers too if possible)
symmetry group
of the packing (Schönfliess notation); if field is empty then the packing has symmetry element C1
reference
for the best known packing so far
records
the sequence of N 's that establish density records

N class radius height density contacts loose boundary symmetry group reference
2 ⟨ 2  0   ⟩ 0.250000000000 0.500000000000 0.785398163397 7 6 D2 [1]
3 ⟨ 4  0   ⟩ 0.166666666667 0.333333333333 0.785398163401 10 8 D2 [1]
4 ⟨ 6  0   ⟩ 0.125000000000 0.250000000000 0.785398163397 13 10 D2 [1]
4 ⟨ 2  0   1⟩ 0.250000000000 1.000000000000 0.785398163397 12 8 D2 [1]
5 ⟨ 8  0   ⟩ 0.100000000000 0.200000000000 0.785398163397 16 12 D2 [1]
6 ⟨ 10  0   ⟩ 0.083333333333 0.166666666667 0.785398163390 19 14 D2 [1]
6 ⟨ 4  0   1⟩ 0.166666666667 0.666666666667 0.785398163400 17 10 D2 [1]
7 ⟨ 12  0   ⟩ 0.071428571428 0.142857142857 0.785398163386 22 16 D2 [1]
8 ⟨ 14  0   ⟩ 0.062500000000 0.125000000000 0.785398163397 25 18 D2 [1]
8 ⟨ 6  0   1⟩ 0.125000000000 0.500000000000 0.785398163397 22 12 D2 [1]
9 ⟨ 16  0   ⟩ 0.055555555555 0.111111111111 0.785398163383 28 20 D2 [1]
9 ⟨ 4  0   2⟩ 0.166666666667 1.000000000000 0.785398163401 24 12 D2 [1]
10 ⟨ 18  0   ⟩ 0.050000000000 0.100000000000 0.785398163397 31 22 D2 [1]
10 ⟨ 8  0   1⟩ 0.100000000000 0.400000000000 0.785398163397 27 14 D2 [1]
11 ⟨ 6  2   ⟩ 0.125000000000 0.683012701892 0.790558119695 32 12 D2 [1]
12 ⟨ 22  0   ⟩ 0.041666666667 0.083333333333 0.785398163413 37 26 D2 [1]
12 ⟨ 10  0   1⟩ 0.083333333333 0.333333333333 0.785398163392 32 16 D2 [1]
12 ⟨ 6  0   2⟩ 0.125000000000 0.750000000000 0.785398163397 31 14 D2 [1]
13 ⟨ 24  0   ⟩ 0.038461538462 0.076923076923 0.785398163417 40 28 D2 [2]
14 ⟨ 8  2   ⟩ 0.100000000000 0.546410161514 0.804931903689 41 14 D2 [2]
15 ⟨ 14  1   ⟩ 0.062500000000 0.233253175473 0.789175513572 44 17 D1 [2]
15 ⟨ 6  2   1⟩ 0.125000000000 0.933012701892 0.789175513572 41 14 D1 [2]
16 ⟨ 15  1   ⟩ 0.058823529412 0.219532400445 0.792270319515 47 18 C2 [2]
17 ⟨ 10  2   ⟩ 0.083333333333 0.455341801261 0.814514426347 50 16 D2 [2]
18 ⟨ 17  1   ⟩ 0.052631578947 0.196423726714 0.797482624231 53 20 C2 [2]
19 ⟨ 18  1   ⟩ 0.050000000000 0.186602540378 0.799697853755 56 21 D1 [2]
19 ⟨ 8  2   1⟩ 0.100000000000 0.746410161514 0.799697853753 52 16 D1 [2]
20 ⟨ 12  2   ⟩ 0.071428571428 0.390292972510 0.821359085384 59 18 D2 [2]
21 ⟨ 13  2   ⟩ 0.066666666667 0.364273441009 0.804931903698 61 17 D1 [2]
22 ⟨ 21  1   ⟩ 0.043478260870 0.162263078590 0.805187770385 65 24 C2 [2]
23 ⟨ 14  2   ⟩ 0.062500000000 0.341506350946 0.826492579681 68 20 D2 [2]
24 ⟨ 15  2   ⟩ 0.058823529412 0.321417742067 0.811696037340 70 19 D1 [2]
25 ⟨ 24  1   ⟩ 0.038461538462 0.143540415676 0.809410783169 74 27 D1 [2]
26 ⟨ 16  2   ⟩ 0.055555555556 0.303561200841 0.830485297471 77 22 D2 [2]
27 ⟨ 17  2   ⟩ 0.052631578947 0.287584295534 0.817036142830 79 21 D1 [2]
28 ⟨ 10  4   ⟩ 0.083333333333 0.744016935856 0.821036738217 81 18 D2 [2]
29 ⟨ 18  2   ⟩ 0.050000000000 0.273205080757 0.833679471678 86 24 D2 [2]
30 ⟨ 19  2   ⟩ 0.047619047619 0.260195315007 0.821359085394 88 23 D1 [2]
31 ⟨ 30  1   ⟩ 0.031250000000 0.116626587737 0.815481364021 92 33 D1 [2]
31 ⟨ 14  2   1⟩ 0.062500000000 0.466506350946 0.815481364025 85 22 D1 [2]
32 ⟨ 20  2   ⟩ 0.045454545455 0.248368255234 0.836292886965 95 26 D2 [2]
33 ⟨ 12  4   ⟩ 0.071428571429 0.637728802163 0.829414664132 96 20 D2 [2]
34 ⟨ 16  3   ⟩ 0.055555555556 0.399786245706 0.824623925162 99 21 D1 [2]
35 ⟨ 22  2   ⟩ 0.041666666667 0.227670900631 0.838470733022 104 28 D2 [2]
36 ⟨ 23  2   ⟩ 0.040000000000 0.218564064606 0.827929958079 106 27 D1 [2]
37 ⟨ 24  2   1⟩ 0.038667951380 0.211285615587 0.822591075363 107 28 D1 [3]
38 ⟨ 24  2   ⟩ 0.038461538462 0.210157754428 0.840313525851 113 30 D2 [2]
39 ⟨ 25  2   ⟩ 0.037037037037 0.202374133894 0.830485297456 115 29 D1 [2]
40 ⟨ 19  3   ⟩ 0.047619047619 0.342673924891 0.831553537963 117 24 C2 [2]
41 ⟨ 26  2   ⟩ 0.035714285714 0.195146486255 0.841893062519 122 32 D2 [2]
42 ⟨ 20  3   ⟩ 0.045454545455 0.327097837396 0.833443432386 123 25 D1 [2]
43 ⟨ 16  4   ⟩ 0.055555555556 0.496011290571 0.840585232003 126 24 D2 [2]
44 ⟨ 28  2   ⟩ 0.033333333333 0.182136720505 0.843261994323 131 34 D2 [2]
45 ⟨ 29  2   ⟩ 0.032258064516 0.176261342424 0.834606812574 133 33 D1 [2]
46 ⟨ 22  3   ⟩ 0.041666666667 0.299839684279 0.836750747592 135 27 D1 [2]
47 ⟨ 30  2   ⟩ 0.031250000000 0.170753175473 0.844459809675 140 36 D2 [2]
48 ⟨ 18  4   ⟩ 0.050000000000 0.446410161514 0.844494930743 141 26 D2 [2]
49 ⟨ 32  2   1⟩ 0.029532317608 0.161367584342 0.832002662133 143 36 D1 [2]
50 ⟨ 32  2   ⟩ 0.029411764706 0.160708871033 0.845516705565 149 38 D2 [2]
51 ⟨ 33  2   ⟩ 0.028571428571 0.156117189004 0.837786267080 151 37 D1 [2]
52 ⟨ 25  3   ⟩ 0.037037037037 0.266524163804 0.840793021718 153 30 C2 [2]
53 ⟨ 20  4   ⟩ 0.045454545455 0.405827419558 0.847693775195 156 28 D2 [2]
54 ⟨ 26  3   ⟩ 0.035714285714 0.257005443668 0.841947957176 159 31 D1 [2]
55 ⟨ 21  4   ⟩ 0.043478260870 0.388182749142 0.841435166519 161 27 D1 [2]
56 ⟨ 36  2   ⟩ 0.026315789474 0.143792147767 0.847296740745 167 42 D2 [2]
57 ⟨ 37  2   ⟩ 0.025641025641 0.140105169619 0.840313525828 169 41 D1 [2]
58 ⟨ 22  4   ⟩ 0.041666666667 0.372008467928 0.850359478888 171 30 D2 [2]
59 ⟨ 38  2   ⟩ 0.025000000000 0.136602540378 0.848053255676 176 44 D2 [2]
60 ⟨ 16  6   ⟩ 0.055555555556 0.688461380301 0.845038565687 175 26 D2 [2]
61 ⟨ 40  2   1⟩ 0.023888463968 0.130528994549 0.837816995337 179 44 D1 [2]
62 ⟨ 40  2   ⟩ 0.023809523810 0.130097657503 0.848737721613 185 46 D2 [2]
63 ⟨ 24  4   ⟩ 0.038461538462 0.343392431934 0.852615074328 186 32 D2 [2]
64 ⟨ 31  3   ⟩ 0.030303030303 0.218065224931 0.846672693197 189 36 C2 [2]
65 ⟨ 42  2   ⟩ 0.022727272727 0.124184127617 0.849359963287 194 48 D2 [2]
66 ⟨ 32  3   ⟩ 0.029411764706 0.211651541844 0.847450885022 195 37 D1 [2]
67 ⟨ 18  6   ⟩ 0.050000000000 0.619615242271 0.849263758502 196 28 D2 [2]
68 ⟨ 26  4   ⟩ 0.035714285714 0.318864401081 0.854548441811 201 34 D2 [2]
69 ⟨ 22  5   ⟩ 0.041666666667 0.444177251577 0.847266758104 202 29 D1 [2]
70 ⟨ 27  4   ⟩ 0.034482758621 0.307869076906 0.849348349901 206 33 D1 [2]
71 ⟨ 46  2   ⟩ 0.020833333333 0.113835450315 0.850448886314 212 52 D2 [2]
72 ⟨ 35  3   ⟩ 0.027027027027 0.194490606019 0.849533073920 213 40 C2 [2]
73 ⟨ 28  4   ⟩ 0.033333333333 0.297606774343 0.856224026985 216 36 D2 [2]
74 ⟨ 20  6   ⟩ 0.045454545455 0.563286583882 0.852720734471 217 30 D2 [2]
75 ⟨ 29  4   ⟩ 0.032258064516 0.288006555815 0.851305373728 221 35 D1 [2]
76 ⟨ 37  3   ⟩ 0.025641025641 0.184516728787 0.850743234995 225 42 C2 [2]
77 ⟨ 50  2   ⟩ 0.019230769231 0.105078877214 0.851370282770 230 56 D2 [2]
78 ⟨ 30  4   ⟩ 0.031250000000 0.279006350946 0.857690164037 231 38 D2 [2]
79 ⟨ 31  4   1⟩ 0.030431016039 0.271694295701 0.845920109914 224 2 35 D1 [2]
80 ⟨ 31  4   ⟩ 0.030303030303 0.270551613039 0.853025182566 236 37 D1 [2]
81 ⟨ 22  6   ⟩ 0.041666666667 0.516346035226 0.855601547758 238 32 D2 [2]
82 ⟨ 40  3   ⟩ 0.023809523810 0.171336962445 0.852342376450 243 45 D1 [2]
83 ⟨ 32  4   ⟩ 0.029411764706 0.262594212655 0.858983814367 246 40 D2 [2]
84 ⟨ 27  5   ⟩ 0.034482758621 0.367594966822 0.853618083130 247 34 C2 [2]
85 ⟨ 33  4   ⟩ 0.028571428571 0.255091520865 0.854548441798 251 39 D1 [2]
86 ⟨ 42  3   ⟩ 0.022727272727 0.163548918698 0.853287323595 255 47 D1 [2]
87 ⟨ 28  5   ⟩ 0.033333333333 0.355341801261 0.854634295102 256 35 D1 [2]
88 ⟨ 34  4   ⟩ 0.027777777778 0.248005645285 0.860133725773 261 42 D2 [2]
89 ⟨ 58  2   ⟩ 0.016666666667 0.091068360252 0.852844517041 266 64 D2 [2]
90 ⟨ 35  4   ⟩ 0.027027027027 0.241302790007 0.855907024402 266 41 D1 [2]
91 ⟨ 25  6   ⟩ 0.037037037037 0.458974253534 0.854427883069 267 33 D1 [2]
92 ⟨ 45  3   ⟩ 0.021276595745 0.153109626015 0.854553954998 273 50 C2 [2]
93 ⟨ 36  4   ⟩ 0.026315789474 0.234952716586 0.861162593872 276 44 D2 [2]
94 ⟨ 46  3   ⟩ 0.020833333333 0.149919842140 0.854940981191 279 51 D1 [2]
95 ⟨ 26  6   ⟩ 0.035714285714 0.442582315908 0.860128540047 280 36 D2 [2]
96 ⟨ 31  5   ⟩ 0.030303030303 0.323038001147 0.857313399489 283 38 C2 [2]
97 ⟨ 38  4   1⟩ 0.025035851747 0.223525172441 0.854515432938 282 43 D1 [2]
98 ⟨ 38  4   ⟩ 0.025000000000 0.223205080757 0.862088575134 291 46 D2 [2]
99 ⟨ 32  5   ⟩ 0.029411764706 0.313536883466 0.858101371373 292 39 D1 [2]
100 ⟨ 39  4   ⟩ 0.024390243902 0.217761054397 0.858226555603 296 45 D1 [2]
101 ⟨ 28  6   1⟩ 0.033354851260 0.413343484887 0.854039910712 289 36 D1 [3]
102 ⟨ 28  6   ⟩ 0.033333333333 0.413076828180 0.861939336971 301 38 D2 [2]
103 ⟨ 40  4   ⟩ 0.023809523810 0.212576267388 0.862926367757 306 48 D2 [2]
104 ⟨ 22  8   ⟩ 0.041666666667 0.660683602523 0.858553178467 305 34 D2 [2]
105 ⟨ 34  5   ⟩ 0.027777777778 0.296118167718 0.859545986483 310 41 D1 [2]
106 ⟨ 52  3   ⟩ 0.018518518519 0.133262081902 0.856962118336 315 57 D1 [2]
107 ⟨ 42  4   1⟩ 0.022756898408 0.203178213877 0.856806304966 312 47 D1 [2]
108 ⟨ 42  4   ⟩ 0.022727272727 0.202913709779 0.863687997331 321 50 D2 [2]
109 ⟨ 30  6   ⟩ 0.031250000000 0.387259526419 0.863523784299 322 40 D2 [2]
110 ⟨ 43  4   ⟩ 0.022222222222 0.198404516228 0.860133725742 326 49 D1 [2]
111 ⟨ 36  5   ⟩ 0.026315789474 0.280533000996 0.860838536845 328 43 D1 [2]
112 ⟨ 31  6   ⟩ 0.030303030303 0.375524389255 0.860402903230 330 39 D1 [2]
113 ⟨ 44  4   ⟩ 0.021739130435 0.194091374571 0.864383398334 336 52 D2 [2]
114 ⟨ 37  5   ⟩ 0.025641025641 0.273339847124 0.861435098526 337 44 C2 [2]
115 ⟨ 45  4   ⟩ 0.021276595745 0.189961770857 0.860965576581 341 51 D1 [2]
116 ⟨ 32  6   ⟩ 0.029411764706 0.364479554277 0.864921826050 343 42 D2 [2]
117 ⟨ 38  5   ⟩ 0.025000000000 0.266506350946 0.862001832145 346 45 D1 [2]
118 ⟨ 46  4   ⟩ 0.020833333333 0.186004233964 0.865020849172 351 54 D2 [2]
119 ⟨ 33  6   ⟩ 0.028571428571 0.354065852726 0.861939336962 351 41 D1 [2]
120 ⟨ 39  5   ⟩ 0.024390243902 0.260006196045 0.862540920189 355 46 C2 [2]
121 ⟨ 26  8   1⟩ 0.035719438916 0.566381941999 0.856318055874 344 35 D1 [2]
122 ⟨ 26  8   ⟩ 0.035714285714 0.566300230734 0.863270503596 359 38 D2 [2]
123 ⟨ 34  6   ⟩ 0.027777777778 0.344230690150 0.866164529831 364 44 D2 [2]
124 ⟨ 30  7   ⟩ 0.031250000000 0.441386114156 0.861892169225 365 39 D1 [2]
125 ⟨ 49  4   ⟩ 0.019607843137 0.175062808437 0.862433548530 371 55 D1 [2]
126 ⟨ 41  5   ⟩ 0.023255813953 0.247912884601 0.863543874742 373 48 C2 [2]
127 ⟨ 22 10   ⟩ 0.041666666667 0.805021169820 0.860446374412 372 36 D2 [2]
128 ⟨ 50  4   ⟩ 0.019230769231 0.171696215967 0.866148646936 381 58 D2 [2]
129 ⟨ 42  5   ⟩ 0.022727272727 0.242278500860 0.864011160404 382 49 D1 [2]
130 ⟨ 36  6   ⟩ 0.026315789474 0.326113285406 0.867276422685 385 46 D2 [2]
131 ⟨ 28  8   ⟩ 0.033333333333 0.528546882018 0.865157433656 386 40 D2 [2]
132 ⟨ 43  5   ⟩ 0.022222222222 0.236894534174 0.864457677804 391 50 C2 [2]
133 ⟨ 52  4   ⟩ 0.018518518519 0.165337096857 0.866649890391 396 60 D2 [2]
134 ⟨ 66  3   ⟩ 0.014705882353 0.105825770922 0.860291049947 399 71 D1 [2]
135 ⟨ 44  5   ⟩ 0.021739130435 0.231744652997 0.864884781432 400 51 D1 [2]
136 ⟨ 33  7   ⟩ 0.028571428571 0.403553018657 0.864275281187 401 42 C2 [2]
137 ⟨ 38  6   ⟩ 0.025000000000 0.309807621135 0.868277126231 406 48 D2 [2]
138 ⟨ 54  4   ⟩ 0.017857142857 0.159432200541 0.867115330658 411 62 D2 [2]
139 ⟨ 30  8   1⟩ 0.031253945349 0.495575260951 0.860725662109 398 39 D1 [2]
140 ⟨ 30  8   ⟩ 0.031250000000 0.495512701892 0.866808497477 413 42 D2 [2]
141 ⟨ 46  5   ⟩ 0.020833333333 0.222088625788 0.865685600632 418 53 D1 [2]
142 ⟨ 56  4   1⟩ 0.017258423685 0.154086714096 0.862333536102 417 61 D1 [2]
143 ⟨ 56  4   ⟩ 0.017241379310 0.153934538453 0.867548671634 426 64 D2 [2]
144 ⟨ 40  6   ⟩ 0.023809523810 0.295054877272 0.869182524728 427 50 D2 [2]
145 ⟨ 57  4   ⟩ 0.016949152542 0.151325478479 0.864772351360 431 63 D1 [2]
146 ⟨ 41  6   1⟩ 0.023286834536 0.288577552452 0.861908368595 420 2 46 D1 [3]
147 ⟨ 41  6   ⟩ 0.023255813953 0.288193135940 0.866655831271 435 49 D1 [2]
148 ⟨ 58  4   ⟩ 0.016666666667 0.148803387171 0.867953123300 441 66 D2 [2]
149 ⟨ 32  8   ⟩ 0.029411764706 0.466364895899 0.868265318487 440 44 D2 [2]
150 ⟨ 49  5   ⟩ 0.019607843137 0.209024588977 0.866769061965 445 56 C2 [2]
151 ⟨ 42  6   ⟩ 0.022727272727 0.281643291941 0.870005614185 448 52 D2 [2]
152 ⟨ 37  7   ⟩ 0.025641025641 0.362162965461 0.866882446465 449 46 C2 [2]
153 ⟨ 60  4   ⟩ 0.016129032258 0.144003277908 0.868331481199 456 68 D2 [2]
154 ⟨ 43  6   ⟩ 0.022222222222 0.275384552120 0.867572927409 456 51 D1 [2]
155 ⟨ 61  4   ⟩ 0.015873015873 0.141717511592 0.865719009687 461 67 D1 [2]
156 ⟨ 38  7   ⟩ 0.025000000000 0.353108891325 0.867452763865 461 47 D1 [2]
157 ⟨ 44  6   1⟩ 0.021748280617 0.269511323275 0.865610198916 457 52 D1 [2]
158 ⟨ 44  6   ⟩ 0.021739130435 0.269397931422 0.870757130734 469 54 D2 [2]
159 ⟨ 52  5   ⟩ 0.018518518519 0.197412111812 0.867732138766 472 59 D1 [2]
160 ⟨ 39  7   ⟩ 0.024390243902 0.344496479341 0.867995260872 473 48 C2 [2]
161 ⟨ 45  6   ⟩ 0.021276595745 0.263666060541 0.868411972835 477 53 D1 [2]
162 ⟨ 53  5   ⟩ 0.018181818182 0.193822800688 0.868029817003 481 60 C2 [2]
163 ⟨ 64  4   ⟩ 0.015151515152 0.135275806519 0.869019404800 486 72 D2 [2]
164 ⟨ 40  7   ⟩ 0.023809523810 0.336294182214 0.868511924784 485 49 D1 [2]
165 ⟨ 46  6   ⟩ 0.020833333333 0.258173017613 0.871446020823 490 56 D2 [2]
166 ⟨ 36  8   1⟩ 0.026318587228 0.417318216551 0.865597041936 479 45 D1 [2]
167 ⟨ 36  8   ⟩ 0.026315789474 0.417273854225 0.870718911772 494 48 D2 [2]
168 ⟨ 66  4   ⟩ 0.014705882353 0.131297106328 0.869333016946 501 74 D2 [2]
169 ⟨ 67  4   1⟩ 0.014521963876 0.129655044790 0.863570626791 494 2 71 D1 [3]
170 ⟨ 33  9   ⟩ 0.028571428571 0.502527350518 0.867566955098 501 44 C2 [2]
171 ⟨ 30 10   ⟩ 0.031250000000 0.603765877365 0.868915334776 504 44 D2 [2]
172 ⟨ 48  6   ⟩ 0.020000000000 0.247846096908 0.872079799777 511 58 D2 [2]
173 ⟨ 68  4   ⟩ 0.014285714286 0.127545760433 0.869628708476 516 76 D2 [2]
174 ⟨ 57  5   ⟩ 0.016949152542 0.180682271828 0.869119622115 517 64 C2 [2]
175 ⟨ 49  6   ⟩ 0.019607843137 0.242986369518 0.869892641112 519 57 D1 [2]
176 ⟨ 38  8   ⟩ 0.025000000000 0.396410161514 0.871761688891 521 50 D2 [2]
177 ⟨ 58  5   ⟩ 0.016666666667 0.177670900631 0.869369369205 526 65 D1 [2]
178 ⟨ 70  4   ⟩ 0.013888888889 0.124002822643 0.869907972653 531 78 D2 [2]
179 ⟨ 50  6   ⟩ 0.019230769231 0.238313554719 0.872664826498 532 60 D2 [2]
180 ⟨ 44  7   ⟩ 0.021739130435 0.307051209847 0.870353943697 533 53 D1 [2]
181 ⟨ 51  6   1⟩ 0.018888338421 0.234070047738 0.866702630308 525 2 56 D1 [2]
182 ⟨ 51  6   ⟩ 0.018867924528 0.233817072555 0.870549163855 540 59 D1 [2]
183 ⟨ 72  4   ⟩ 0.013513513514 0.120651395004 0.870172141536 546 80 D2 [2]
184 ⟨ 45  7   ⟩ 0.021276595745 0.300518205383 0.870765458570 545 54 C2 [2]
185 ⟨ 40  8   ⟩ 0.023809523810 0.377533487156 0.872705153957 548 52 D2 [2]
186 ⟨ 52  6   ⟩ 0.018518518519 0.229487126767 0.873206517908 553 62 D2 [2]
187 ⟨ 33 10   ⟩ 0.028571428571 0.552014516448 0.868770152572 551 45 D1 [2]
188 ⟨ 46  7   ⟩ 0.020833333333 0.294257409437 0.871159826932 557 55 D1 [2]
189 ⟨ 53  6   ⟩ 0.018181818182 0.225314633553 0.871157939540 561 61 D1 [2]
190 ⟨ 37  9   ⟩ 0.025641025641 0.450986083798 0.870184050010 561 48 C2 [2]
191 ⟨ 47  7   1⟩ 0.020416227109 0.288366052773 0.867341413605 554 1 54 [2]
192 ⟨ 47  7   ⟩ 0.020408163265 0.288252156183 0.871538098678 569 56 C2 [2]
193 ⟨ 54  6   ⟩ 0.017857142857 0.221291157954 0.873709516995 574 64 D2 [2]
194 ⟨ 42  8   ⟩ 0.022727272727 0.360372874103 0.873562849386 575 54 D2 [2]
195 ⟨ 38  9   ⟩ 0.025000000000 0.439711431703 0.870756539518 576 49 D1 [2]
196 ⟨ 48  7   ⟩ 0.020000000000 0.282487113060 0.871901239577 581 57 D1 [2]
197 ⟨ 43  8   1⟩ 0.022231783688 0.352516198496 0.867732047078 568 2 51 D1 [2]
198 ⟨ 43  8   ⟩ 0.022222222222 0.352364588012 0.871761688874 586 53 D1 [2]
199 ⟨ 56  6   1⟩ 0.017247134394 0.213731747117 0.870097274518 583 64 D1 [2]
200 ⟨ 56  6   ⟩ 0.017241379310 0.213660428369 0.874177826525 595 66 D2 [2]
201 ⟨ 66  5   ⟩ 0.014705882353 0.156768441733 0.871102907303 598 73 D1 [2]
202 ⟨ 30 12   ⟩ 0.031250000000 0.712019052838 0.870381538016 595 46 D2 [2]
203 ⟨ 44  8   ⟩ 0.021739130435 0.344704488273 0.874345962690 602 56 D2 [2]
204 ⟨ 36 10   ⟩ 0.026315789474 0.508434423044 0.872926947608 603 50 D2 [2]
205 ⟨ 40  9   ⟩ 0.023809523810 0.418772792098 0.871819734351 606 51 D1 [2]
206 ⟨ 58  6   1⟩ 0.016672044413 0.206605056757 0.870670566775 604 66 D1 [2]
207 ⟨ 58  6   ⟩ 0.016666666667 0.206538414090 0.874614915509 616 68 D2 [2]
208 ⟨ 51  7   ⟩ 0.018867924528 0.266497276471 0.872908441599 617 60 C2 [2]
209 ⟨ 37 10   ⟩ 0.025641025641 0.495397642966 0.871390877039 617 49 D1 [2]
210 ⟨ 59  6   ⟩ 0.016393442623 0.203152538449 0.872744748162 624 67 D1 [2]
211 ⟨ 46  8   1⟩ 0.020835086748 0.330369604118 0.871009458374 614 55 D1 [2]
212 ⟨ 46  8   ⟩ 0.020833333333 0.330341801261 0.875063816474 629 58 D2 [2]
213 ⟨ 70  5   ⟩ 0.013888888889 0.148059083859 0.871825214861 634 77 D1 [2]
214 ⟨ 60  6   ⟩ 0.016129032258 0.199875884603 0.875023805096 637 70 D2 [4]
215 ⟨ 38 10   ⟩ 0.025000000000 0.483012701892 0.873996711003 636 52 D2 [4]
216 ⟨ 53  7   ⟩ 0.018181818182 0.256806466418 0.873518867126 641 62 C2 [4]
217 ⟨ 61  6   ⟩ 0.015873015873 0.196703251514 0.873206517864 645 69 D1 [4]
218 ⟨ 86  4   ⟩ 0.011363636364 0.101456854889 0.871685108498 651 94 D2 [4]
219 ⟨ 72  5   ⟩ 0.013513513514 0.144057486998 0.872157085947 652 79 D1 [4]
220 ⟨ 54  7   ⟩ 0.017857142857 0.252220636660 0.873807729156 653 63 D1 [4]
221 ⟨ 48  8   ⟩ 0.020000000000 0.317128129211 0.875724242023 656 60 D2 [4]
222 ⟨ 73  5   ⟩ 0.013333333333 0.142136720505 0.872316383936 661 80 C2 [4]
223 ⟨ 88  4   ⟩ 0.011111111111 0.099202258114 0.871862822002 666 96 D2 [4]
224 ⟨ 55  7   ⟩ 0.017543859649 0.247795713210 0.874086455706 665 64 C2 [4]
225 ⟨ 49  8   ⟩ 0.019607843137 0.310909930599 0.874092602475 667 59 D1 [4]
226 ⟨ 40 10   ⟩ 0.023809523810 0.460012097040 0.874964592224 669 54 D2 [4]
227 ⟨ 64  6   1⟩ 0.015155959439 0.187817269596 0.872181912359 667 72 D1 [4]
228 ⟨ 64  6   ⟩ 0.015151515152 0.187762194627 0.875767240848 679 74 D2 [4]
229 ⟨ 50  8   1⟩ 0.019232263255 0.304954583326 0.872591504066 668 59 D1 [4]
230 ⟨ 50  8   ⟩ 0.019230769231 0.304930893472 0.876333865603 683 62 D2 [4]
231 ⟨ 65  6   ⟩ 0.014925373134 0.184959773812 0.874047352219 687 73 D1 [4]
232 ⟨ 57  7   ⟩ 0.016949152542 0.239395858525 0.874615563703 689 66 C2 [4]
233 ⟨ 92  4   ⟩ 0.010638297872 0.094980885428 0.872195562285 696 100 D2 [4]
234 ⟨ 51  8   ⟩ 0.018867924528 0.299177480388 0.874752294999 694 61 D1 [4]
235 ⟨ 66  6   ⟩ 0.014705882353 0.182239777138 0.876106160010 700 76 D2 [4]
236 ⟨ 58  7   ⟩ 0.016666666667 0.235405927550 0.874866890086 701 67 D1 [4]
237 ⟨ 42 10   ⟩ 0.022727272727 0.439102456266 0.875844484154 702 56 D2 [4]
238 ⟨ 67  6   ⟩ 0.014492753623 0.179598620948 0.874431211414 708 75 D1 [4]
239 ⟨ 52  8   ⟩ 0.018518518519 0.293637156677 0.876898331884 710 64 D2 [4]
240 ⟨ 59  7   ⟩ 0.016393442623 0.231546813983 0.875109976163 713 68 C2 [4]
241 ⟨ 36 12   ⟩ 0.026315789474 0.599594991864 0.874463577558 712 52 D2 [4]
242 ⟨ 68  6   ⟩ 0.014285714286 0.177032926363 0.876425712435 721 78 D2 [4]
243 ⟨ 53  8   ⟩ 0.018181818182 0.288298299283 0.875364009936 721 63 D1 [4]
244 ⟨ 60  7   ⟩ 0.016129032258 0.227812187951 0.875345220768 725 69 D1 [4]
245 ⟨ 48  9   ⟩ 0.020000000000 0.351769145362 0.875221957670 726 59 D1 [4]
246 ⟨ 40 11   ⟩ 0.023809523810 0.501251401982 0.874038974255 727 53 D1 [4]
247 ⟨ 54  8   1⟩ 0.017858431065 0.283170541710 0.873947526937 722 63 D1 [4]
248 ⟨ 54  8   ⟩ 0.017857142857 0.283150115367 0.877422479065 737 66 D2 [4]
249 ⟨ 70  6   ⟩ 0.013888888889 0.172115345075 0.876727511902 742 80 D2 [4]
250 ⟨ 49  9   ⟩ 0.019607843137 0.344871711140 0.875572186520 741 60 C2 [4]
251 ⟨ 55  8   1⟩ 0.017549818472 0.278277054993 0.872753206187 730 2 63 D1 [4]
252 ⟨ 55  8   ⟩ 0.017543859649 0.278182569483 0.875932797439 748 65 D1 [4]
253 ⟨ 45 10   ⟩ 0.021276595745 0.411074639908 0.875294083683 749 57 D1 [4]
254 ⟨ 38 12   ⟩ 0.025000000000 0.569615242271 0.875552121409 751 54 D2 [4]
255 ⟨ 50  9   ⟩ 0.019230769231 0.338239562848 0.875908945100 756 61 D1 [4]
256 ⟨ 72  6   ⟩ 0.013513513514 0.167463578992 0.877012997952 763 82 D2 [4]
257 ⟨ 56  8   ⟩ 0.017241379310 0.273386318285 0.877910478199 764 68 D2 [4]
258 ⟨ 42 11   ⟩ 0.022727272727 0.478467247347 0.875007975279 763 55 D1 [4]
259 ⟨ 46 10   ⟩ 0.020833333333 0.402510584910 0.877384295126 768 60 D2 [4]
260 ⟨ 51  9   ⟩ 0.018867924528 0.331857684304 0.876232995714 771 62 C2 [4]
261 ⟨ 57  8   ⟩ 0.016949152542 0.268752651874 0.876463023075 775 67 D1 [4]
262 ⟨ 74  6   1⟩ 0.013161246290 0.163098176169 0.874170390041 772 82 D1 [4]
263 ⟨ 74  6   ⟩ 0.013157894737 0.163056642703 0.877283458332 784 84 D2 [4]
264 ⟨ 65  7   ⟩ 0.014925373134 0.210811278403 0.876416110410 785 74 C2 [4]
265 ⟨ 52  9   ⟩ 0.018518518519 0.325712171632 0.876545044574 786 63 D1 [4]
266 ⟨ 58  8   ⟩ 0.016666666667 0.264273441009 0.878365944146 791 70 D2 [4]
267 ⟨ 40 12   ⟩ 0.023809523810 0.542490706924 0.876536994473 790 56 D2 [4]
268 ⟨ 66  7   ⟩ 0.014705882353 0.207711112544 0.876611390338 797 75 D1 [4]
269 ⟨ 48 10   1⟩ 0.020000209583 0.386414210761 0.874818898467 783 60 D1 [4]
270 ⟨ 48 10   ⟩ 0.020000000000 0.386410161514 0.878061811983 801 62 D2 [4]
271 ⟨ 67  7   1⟩ 0.014496819801 0.204758238700 0.873822573204 794 1 74 [4]
272 ⟨ 67  7   ⟩ 0.014492753623 0.204700806565 0.876801009904 809 76 C2 [4]
273 ⟨ 77  6   ⟩ 0.012658227848 0.156864618296 0.876058968706 813 85 D1 [4]
274 ⟨ 60  8   1⟩ 0.016130083192 0.255765155337 0.875653469748 803 69 D1 [4]
275 ⟨ 60  8   ⟩ 0.016129032258 0.255748491299 0.878792025086 818 72 D2 [4]
276 ⟨ 68  7   ⟩ 0.014285714286 0.201776509328 0.876985211857 821 77 D1 [4]
277 ⟨ 78  6   ⟩ 0.012500000000 0.154903810568 0.877783810092 826 88 D2 [4]
278 ⟨ 36 14   ⟩ 0.026315789474 0.690755560683 0.875594622471 821 54 D2 [4]
279 ⟨ 61  8   ⟩ 0.015873015873 0.251688991437 0.877422479078 829 71 D1 [4]
280 ⟨ 42 12   ⟩ 0.022727272727 0.517832038428 0.877432333532 829 58 D2 [4]
281 ⟨ 50 10   ⟩ 0.019230769231 0.371548232225 0.878687212155 834 64 D2 [4]
282 ⟨ 46 11   ⟩ 0.020833333333 0.438594976735 0.876703727161 835 59 D1 [4]
283 ⟨ 62  8   1⟩ 0.015625986275 0.247771989724 0.876151031941 830 71 D1 [4]
284 ⟨ 62  8   ⟩ 0.015625000000 0.247756350946 0.879191476013 845 74 D2 [4]
285 ⟨ 56  9   ⟩ 0.017241379310 0.303249263243 0.877685636616 846 67 D1 [4]
286 ⟨ 51 10   ⟩ 0.018867924528 0.364537888221 0.877448211806 848 63 D1 [4]
287 ⟨ 81  6   ⟩ 0.012048192771 0.149304877656 0.876600247028 855 89 D1 [4]
288 ⟨ 63  8   ⟩ 0.015384615385 0.243944714778 0.877857924521 856 73 D1 [4]
289 ⟨ 57  9   1⟩ 0.016950313072 0.298129857150 0.874979124261 843 1 65 [4]
290 ⟨ 57  9   ⟩ 0.016949152542 0.298109445222 0.877946619555 861 68 C2 [4]
291 ⟨ 82  6   ⟩ 0.011904761905 0.147527438636 0.878236509360 868 92 D2 [4]
292 ⟨ 52 10   ⟩ 0.018518518519 0.357787186587 0.879266286394 867 66 D2 [4]
293 ⟨ 64  8   ⟩ 0.015151515152 0.240248582736 0.879566717841 872 76 D2 [4]
294 ⟨ 48 11   ⟩ 0.020000000000 0.421051177665 0.877449858022 871 61 D1 [4]
295 ⟨ 58  9   ⟩ 0.016666666667 0.293140954469 0.878198903138 876 69 D1 [4]
296 ⟨ 73  7   ⟩ 0.013333333333 0.188324742040 0.877832540482 881 82 C2 [4]
297 ⟨ 65  8   ⟩ 0.014925373134 0.236662782993 0.878267373099 883 75 D1 [4]
298 ⟨ 84  6   ⟩ 0.011627906977 0.144096567970 0.878447067146 889 94 D2 [4]
299 ⟨ 45 12   ⟩ 0.021276595745 0.484778929592 0.877165636822 885 59 D1 [4]
300 ⟨ 59  9   ⟩ 0.016393442623 0.288335365051 0.878442915030 891 70 C2 [4]
301 ⟨ 85  6   ⟩ 0.011494252874 0.142440285579 0.877091752718 897 93 D1 [4]
302 ⟨ 66  8   ⟩ 0.014705882353 0.233182447949 0.879919886523 899 78 D2 [4]
303 ⟨ 54 10   ⟩ 0.017857142857 0.345009072780 0.879803998105 900 68 D2 [4]
304 ⟨ 75  7   ⟩ 0.012987012987 0.183433190298 0.878140660058 905 84 C2 [4]
305 ⟨ 60  9   ⟩ 0.016129032258 0.283684794647 0.878679055586 906 71 D1 [4]
306 ⟨ 46 12   ⟩ 0.020833333333 0.474679368559 0.878999176977 907 62 D2 [4]
307 ⟨ 55 10   1⟩ 0.017545103051 0.338980305192 0.875841719938 893 2 64 D1 [4]
308 ⟨ 55 10   ⟩ 0.017543859649 0.338956282030 0.878632352459 914 67 D1 [4]
309 ⟨ 61  9   1⟩ 0.015874033708 0.279199763543 0.876128690444 903 1 69 [4]
310 ⟨ 61  9   ⟩ 0.015873015873 0.279181861399 0.878907699633 921 72 C2 [4]
311 ⟨ 68  8   ⟩ 0.014285714286 0.226520092294 0.880252874206 926 80 D2 [4]
312 ⟨ 88  6   ⟩ 0.011111111111 0.137692276060 0.878840108285 931 98 D2 [4]
313 ⟨ 56 10   1⟩ 0.017241535064 0.333115217447 0.877509035477 915 68 D1 [4]
314 ⟨ 56 10   ⟩ 0.017241379310 0.333112208202 0.880304626282 933 70 D2 [4]
315 ⟨ 62  9   ⟩ 0.015625000000 0.274819644814 0.879129198553 936 73 D1 [4]
316 ⟨ 78  7   ⟩ 0.012500000000 0.176554445662 0.878573953147 941 87 D1 [4]
317 ⟨ 52 11   1⟩ 0.018518698202 0.389865984336 0.876021365188 922 1 63 [4]
318 ⟨ 52 11   ⟩ 0.018518518519 0.389862201542 0.878776312879 943 65 D1 [4]
319 ⟨ 48 12   ⟩ 0.020000000000 0.455692193817 0.879688588124 946 64 D2 [4]
320 ⟨ 70  8   ⟩ 0.013888888889 0.220227867508 0.880567362529 953 82 D2 [4]
321 ⟨106  5   ⟩ 0.009259259259 0.098706055906 0.875918290924 958 113 D1 [4]
322 ⟨ 45 13   ⟩ 0.021276595745 0.521631074434 0.877903081416 953 60 C2 [4]
323 ⟨ 42 14   ⟩ 0.022727272727 0.596561620590 0.878601078784 956 60 D2 [4]
324 ⟨ 71  8   ⟩ 0.013698630137 0.217211047405 0.879361105856 964 81 D1 [4]
325 ⟨ 58 10   ⟩ 0.016666666667 0.322008467928 0.880771879340 966 72 D2 [4]
326 ⟨ 92  6   ⟩ 0.010638297872 0.131833030270 0.879199699126 973 102 D2 [4]
327 ⟨ 81  7   1⟩ 0.012051002780 0.170212649246 0.876500498357 962 1 88 [4]
328 ⟨ 81  7   ⟩ 0.012048192771 0.170172959674 0.878975923833 977 90 C2 [4]
329 ⟨ 72  8   ⟩ 0.013513513514 0.214275762980 0.880864851555 980 84 D2 [4]
330 ⟨ 65  9   ⟩ 0.014925373134 0.262514287584 0.879754023814 981 76 C2 [4]
331 ⟨ 50 12   1⟩ 0.019230777756 0.438165765227 0.877673775464 964 63 D1 [4]
332 ⟨ 50 12   ⟩ 0.019230769231 0.438165570977 0.880324967642 985 66 D2 [4]
333 ⟨ 73  8   ⟩ 0.013333333333 0.211418752807 0.879686795111 991 83 D1 [4]
334 ⟨ 66  9   1⟩ 0.014706756005 0.258669149548 0.877375451235 978 1 74 [4]
335 ⟨ 66  9   ⟩ 0.014705882353 0.258653783355 0.879950047483 996 77 D1 [4]
336 ⟨ 60 10   ⟩ 0.016129032258 0.311621097995 0.881208986930 999 74 D2 [4]
337 ⟨ 74  8   1⟩ 0.013158594138 0.208648017105 0.878586438616 992 83 D1 [4]
338 ⟨ 74  8   ⟩ 0.013157894737 0.208636927113 0.881146683169 1007 86 D2 [4]
339 ⟨ 67  9   1⟩ 0.014493602135 0.254920101810 0.877603119360 993 1 75 [4]
340 ⟨ 67  9   ⟩ 0.014492753623 0.254905177799 0.880140389233 1011 78 C2 [4]
341 ⟨ 61 10   ⟩ 0.015873015873 0.306674731360 0.880126625179 1013 73 D1 [4]
342 ⟨ 75  8   ⟩ 0.012987012987 0.205927356631 0.879995565524 1018 85 D1 [4]
343 ⟨ 48 13   ⟩ 0.020000000000 0.490333209968 0.879048172366 1016 63 D1 [4]
344 ⟨ 85  7   ⟩ 0.011494252874 0.162348915552 0.879468761543 1025 94 C2 [4]
345 ⟨ 52 12   ⟩ 0.018518518519 0.421937216497 0.880914207938 1024 68 D2 [4]
346 ⟨ 62 10   1⟩ 0.015625127919 0.301885410141 0.879085284085 1014 74 D1 [4]
347 ⟨ 62 10   ⟩ 0.015625000000 0.301882938683 0.881618775341 1032 76 D2 [4]
348 ⟨ 57 11   ⟩ 0.016949152542 0.356823031920 0.880181455603 1033 70 C2 [4]
349 ⟨ 40 16   ⟩ 0.023809523810 0.707447926693 0.878581877293 1032 60 D2 [4]
350 ⟨ 69  9   ⟩ 0.014084507042 0.247724750255 0.880504987564 1041 80 C2 [4]
351 ⟨ 77  8   ⟩ 0.012658227848 0.200714005830 0.880288701943 1045 87 D1 [4]
352 ⟨ 63 10   ⟩ 0.015384615385 0.297238585780 0.880563412631 1046 75 D1 [4]
353 ⟨ 46 14   ⟩ 0.020833333333 0.546848152208 0.880187819631 1046 64 D2 [4]
354 ⟨ 58 11   ⟩ 0.016666666667 0.350875981388 0.880434381383 1051 71 D1 [4]
355 ⟨ 70  9   ⟩ 0.013888888889 0.244284128724 0.880679690979 1056 81 D1 [4]
356 ⟨ 78  8   ⟩ 0.012500000000 0.198205080757 0.881668071719 1061 90 D2 [4]
357 ⟨ 50 13   ⟩ 0.019230769231 0.471474240354 0.879738163103 1058 65 D1 [4]
358 ⟨ 64 10   ⟩ 0.015151515152 0.292734970844 0.882003728143 1065 78 D2 [4]
359 ⟨ 71  9   1⟩ 0.013699388206 0.240951104065 0.878451413426 1053 1 79 [4]
360 ⟨ 71  9   ⟩ 0.013698630137 0.240937770796 0.880849607945 1071 82 C2 [4]
361 ⟨102  6   ⟩ 0.009615384615 0.119156777360 0.879977660144 1078 112 D2 [4]
362 ⟨ 65 10   1⟩ 0.014926273061 0.288383179216 0.878600169252 1058 2 74 D1 [4]
363 ⟨ 65 10   ⟩ 0.014925373134 0.288365792174 0.880974123097 1079 77 D1 [4]
364 ⟨ 55 12   ⟩ 0.017543859649 0.399729994576 0.880511043483 1080 69 D1 [4]
365 ⟨ 80  8   ⟩ 0.012195121951 0.193370810495 0.881909690779 1088 92 D2 [4]
366 ⟨ 60 11   ⟩ 0.016129032258 0.339557401343 0.880915756071 1087 73 D1 [4]
367 ⟨ 48 14   1⟩ 0.020000000151 0.524974230074 0.878492280366 1067 64 D1 [4]
368 ⟨ 48 14   ⟩ 0.020000000000 0.524974226119 0.880885985636 1091 66 D2 [4]
369 ⟨ 66 10   ⟩ 0.014705882353 0.284125118760 0.882366036560 1098 80 D2 [4]
370 ⟨ 73  9   ⟩ 0.013333333333 0.234512763575 0.881175848494 1101 84 C2 [4]
371 ⟨ 56 12   ⟩ 0.017241379310 0.392838098118 0.881970776622 1102 72 D2 [4]
372 ⟨ 61 11   ⟩ 0.015873015873 0.334167601322 0.881144982137 1105 74 C2 [4]
373 ⟨ 82  8   1⟩ 0.011905334428 0.188775821734 0.879823452073 1100 91 D1 [4]
374 ⟨ 82  8   ⟩ 0.011904761905 0.188766743578 0.882139804270 1115 94 D2 [4]
375 ⟨ 74  9   ⟩ 0.013157894737 0.231427069317 0.881332529899 1116 85 D1 [4]
376 ⟨ 93  7   ⟩ 0.010526315789 0.148677427926 0.880329930322 1121 102 C2 [4]
377 ⟨ 57 12   ⟩ 0.016949152542 0.386179825268 0.881044040457 1119 71 D1 [4]
378 ⟨ 62 11   ⟩ 0.015625000000 0.328946232551 0.881367044888 1123 75 D1 [4]
379 ⟨ 68 10   1⟩ 0.014285821215 0.276009324160 0.880391316529 1113 80 D1 [4]
380 ⟨ 68 10   ⟩ 0.014285714286 0.276007258224 0.882707641712 1131 82 D2 [4]
381 ⟨108  6   1⟩ 0.009092508847 0.112677141439 0.878226408538 1129 116 D1 [4]
382 ⟨108  6   ⟩ 0.009090909091 0.112657316776 0.880376542078 1141 118 D2 [4]
383 ⟨ 84  8   ⟩ 0.011627906977 0.184376819309 0.882359214745 1142 96 D2 [4]
384 ⟨ 58 12   ⟩ 0.016666666667 0.379743494847 0.882446232637 1141 74 D2 [4]
385 ⟨ 69 10   ⟩ 0.014084507042 0.272119832052 0.881726128329 1145 81 D1 [4]
386 ⟨ 85  8   1⟩ 0.011496810404 0.182298098763 0.879244735899 1135 2 93 D1 [4]
387 ⟨ 85  8   ⟩ 0.011494252874 0.182257545524 0.881326472377 1153 95 D1 [4]
388 ⟨ 96  7   ⟩ 0.010204081633 0.144126078092 0.880616620623 1157 105 D1 [4]
389 ⟨110  6   ⟩ 0.008928571429 0.110645578977 0.880500005568 1162 120 D2 [4]
390 ⟨ 64 11   ⟩ 0.015151515152 0.318978164898 0.881790982916 1159 77 D1 [4]
391 ⟨ 70 10   ⟩ 0.013888888889 0.268340389940 0.883030268753 1164 84 D2 [4]
392 ⟨ 86  8   ⟩ 0.011363636364 0.180186437052 0.882568652032 1169 98 D2 [4]
393 ⟨ 78  9   2⟩ 0.012500000000 0.219855715851 0.877454666747 1168 87 [4]
394 ⟨ 78  9   1⟩ 0.012500631208 0.219866817827 0.879731797075 1158 1 86 [4]
395 ⟨ 78  9   ⟩ 0.012500000000 0.219855715851 0.881920084899 1176 89 D1 [4]
396 ⟨ 71 10   ⟩ 0.013698630137 0.264664494188 0.882071226650 1178 83 D1 [4]
397 ⟨ 60 12   ⟩ 0.016129032258 0.367493704691 0.882891013961 1180 76 D2 [4]
398 ⟨ 52 14   ⟩ 0.018518518519 0.486087246407 0.882127169638 1181 70 D2 [4]
399 ⟨ 56 13   ⟩ 0.017241379310 0.422701043076 0.881522621795 1184 71 D1 [4]
400 ⟨ 79  9   ⟩ 0.012345679012 0.217141447755 0.882057906393 1191 90 C2 [4]
401 ⟨ 88  8   ⟩ 0.011111111111 0.176182294006 0.882768780906 1196 100 D2 [4]
402 ⟨ 72 10   ⟩ 0.013513513514 0.261087946969 0.883335456563 1197 86 D2 [4]
403 ⟨ 61 12   ⟩ 0.015873015873 0.361660471283 0.882008511255 1197 75 D1 [4]
404 ⟨100  7   ⟩ 0.009803921569 0.138474075029 0.880972641098 1205 109 D1 [4]
405 ⟨ 80  9   ⟩ 0.012195121951 0.214493381319 0.882192366457 1206 91 D1 [4]
406 ⟨ 57 13   ⟩ 0.016949152542 0.415536618617 0.881784745675 1205 72 C2 [4]
407 ⟨ 73 10   ⟩ 0.013333333333 0.257606774343 0.882397919651 1211 85 D1 [4]
408 ⟨ 67 11   ⟩ 0.014492753623 0.305109549033 0.882380809581 1213 80 C2 [4]
409 ⟨ 62 12   1⟩ 0.015625005628 0.356009654654 0.881153904130 1198 75 D1 [4]
410 ⟨ 62 12   ⟩ 0.015625000000 0.356009526419 0.883307996501 1219 78 D2 [4]
411 ⟨ 74 10   2⟩ 0.013157894737 0.254217211522 0.879345528105 1222 86 D1 [4]
412 ⟨ 74 10   1⟩ 0.013157985449 0.254218964133 0.881491131963 1212 86 D1 [4]
413 ⟨ 74 10   ⟩ 0.013157894737 0.254217211522 0.883624581770 1230 88 D2 [4]
414 ⟨ 68 11   ⟩ 0.014285714286 0.300750841189 0.882566183761 1231 81 D1 [4]
415 ⟨ 82  9   ⟩ 0.011904761905 0.209386396049 0.882451682331 1236 93 D1 [4]
416 ⟨ 63 12   ⟩ 0.015384615385 0.350532456782 0.882446232646 1236 77 D1 [4]
417 ⟨ 48 16   ⟩ 0.020000000000 0.594256258422 0.881804183283 1236 68 D2 [4]
418 ⟨ 75 10   ⟩ 0.012987012987 0.250915689295 0.882707641674 1244 87 D1 [4]
419 ⟨ 92  8   ⟩ 0.010638297872 0.168685175112 0.883143490383 1250 104 D2 [4]
420 ⟨ 69 11   ⟩ 0.014084507042 0.296514913849 0.882746336005 1249 82 C2 [4]
421 ⟨ 64 12   2⟩ 0.015151515152 0.345221358952 0.879521458213 1248 80 D1 [4]
422 ⟨ 64 12   1⟩ 0.015151520444 0.345221479533 0.881610890726 1237 77 D1 [4]
423 ⟨ 64 12   ⟩ 0.015151515152 0.345221358952 0.883699707420 1258 80 D2 [4]
424 ⟨ 76 10   ⟩ 0.012820512821 0.247698821483 0.883898880127 1263 90 D2 [4]
425 ⟨ 84  9   ⟩ 0.011627906977 0.204516944978 0.882698936939 1266 95 D1 [4]
426 ⟨ 70 11   ⟩ 0.013888888889 0.292396651156 0.882921484136 1267 83 D1 [4]
427 ⟨ 60 13   ⟩ 0.016129032258 0.395430008039 0.882520383704 1268 75 D1 [4]
428 ⟨ 94  8   ⟩ 0.010416666667 0.165170900631 0.883319135579 1277 106 D2 [4]
429 ⟨ 77 10   ⟩ 0.012658227848 0.244563393363 0.883001681517 1277 89 D1 [4]
430 ⟨ 85  9   ⟩ 0.011494252874 0.202166175496 0.882818301257 1281 96 C2 [4]
431 ⟨122  6   ⟩ 0.008064516129 0.099937942302 0.881157149520 1288 132 D2 [4]
432 ⟨ 71 11   ⟩ 0.013698630137 0.288391217579 0.883091833627 1285 84 C2 [4]
433 ⟨ 61 13   1⟩ 0.015873021681 0.389153483643 0.880716363323 1265 1 73 [4]
434 ⟨ 61 13   ⟩ 0.015873015873 0.389153341244 0.882750027318 1289 76 C2 [4]
435 ⟨ 78 10   ⟩ 0.012500000000 0.241506350946 0.884159463456 1296 92 D2 [4]
436 ⟨ 66 12   ⟩ 0.014705882353 0.335067789571 0.884068376419 1297 82 D2 [4]
437 ⟨ 96  8   ⟩ 0.010204081633 0.161800065924 0.883487611483 1304 108 D2 [4]
438 ⟨ 72 11   ⟩ 0.013513513514 0.284494038963 0.883257579153 1303 85 D1 [4]
439 ⟨ 79 10   1⟩ 0.012346294731 0.238536687052 0.881317695586 1289 2 88 D1 [4]
440 ⟨ 79 10   ⟩ 0.012345679012 0.238524791058 0.883281200848 1310 91 D1 [4]
441 ⟨ 62 13   ⟩ 0.015625000000 0.383072820287 0.882972494565 1310 77 D1 [4]
442 ⟨ 67 12   ⟩ 0.014492753623 0.330211734650 0.883245549818 1314 81 D1 [4]
443 ⟨ 58 14   ⟩ 0.016666666667 0.437478521766 0.883678649575 1316 76 D2 [4]
444 ⟨ 73 11   ⟩ 0.013333333333 0.280700785110 0.883418904631 1321 86 C2 [4]
445 ⟨ 88  9   ⟩ 0.011111111111 0.195427302979 0.883160478810 1326 99 D1 [4]
446 ⟨ 80 10   ⟩ 0.012195121951 0.235615952143 0.884407335433 1329 94 D2 [4]
447 ⟨ 46 18   ⟩ 0.020833333333 0.691185719505 0.881820446373 1324 68 D2 [4]
448 ⟨ 63 13   ⟩ 0.015384615385 0.377179392283 0.883188116707 1331 78 C2 [4]
449 ⟨ 68 12   ⟩ 0.014285714286 0.325494424155 0.884415978691 1336 84 D2 [4]
450 ⟨ 74 11   ⟩ 0.013157894737 0.277007353727 0.883575984871 1339 87 D1 [4]
451 ⟨ 81 10   ⟩ 0.012048192771 0.232777205731 0.883547249441 1343 93 D1 [4]
452 ⟨112  7   ⟩ 0.008771929825 0.123897856605 0.881890799161 1349 121 D1 [4]
453 ⟨100  8   2⟩ 0.009803921569 0.155454965300 0.879919886579 1348 112 D1 [4]
454 ⟨100  8   1⟩ 0.009804309851 0.155461122059 0.881897240523 1343 109 D1 [4]
455 ⟨100  8   ⟩ 0.009803921569 0.155454965300 0.883804742590 1358 112 D2 [4]
456 ⟨ 75 11   ⟩ 0.012987012987 0.273409855627 0.883728985018 1357 88 C2 [4]
457 ⟨ 82 10   ⟩ 0.011904761905 0.230006048520 0.884643404085 1362 96 D2 [4]
458 ⟨ 60 14   ⟩ 0.016129032258 0.423366311387 0.884129079193 1361 78 D2 [4]
459 ⟨101  8   ⟩ 0.009708737864 0.153945693792 0.882918391867 1369 111 D1 [4]
460 ⟨ 91  9   ⟩ 0.010752688172 0.189123196431 0.883480580482 1371 102 C2 [4]
461 ⟨ 70 12   1⟩ 0.013888893336 0.316453013694 0.882829521466 1354 83 D1 [4]
462 ⟨ 70 12   ⟩ 0.013888888889 0.316452912373 0.884744269679 1375 86 D2 [4]
463 ⟨102  8   1⟩ 0.009615758107 0.152471368970 0.882083347995 1370 111 D1 [4]
464 ⟨102  8   ⟩ 0.009615384615 0.152465446736 0.883954159994 1385 114 D2 [4]
465 ⟨ 92  9   ⟩ 0.010638297872 0.187111247533 0.883582740536 1386 103 D1 [4]
466 ⟨ 48 18   ⟩ 0.020000000000 0.663538290725 0.882530637364 1381 70 D2 [4]
467 ⟨ 84 10   1⟩ 0.011627977820 0.224658439370 0.882983127634 1377 96 D1 [4]
468 ⟨ 84 10   ⟩ 0.011627906977 0.224657070648 0.884868492734 1395 98 D2 [4]
469 ⟨ 66 13   ⟩ 0.014705882353 0.360539124976 0.883796932008 1394 81 D1 [4]
470 ⟨ 93  9   ⟩ 0.010526315789 0.185141655454 0.883682749876 1401 104 C2 [4]
471 ⟨ 58 15   1⟩ 0.016666666771 0.466346038153 0.881373508862 1374 1 73 [4]
472 ⟨ 58 15   ⟩ 0.016666666667 0.466346035226 0.883244784718 1401 75 D1 [4]
473 ⟨ 62 14   ⟩ 0.015625000000 0.410136114156 0.884551357009 1406 80 D2 [4]
474 ⟨ 78 11   ⟩ 0.012500000000 0.263156986041 0.884165035506 1411 91 D1 [4]
475 ⟨ 72 12   ⟩ 0.013513513514 0.307900130957 0.885054815279 1414 88 D2 [4]
476 ⟨ 67 13   ⟩ 0.014492753623 0.355313920267 0.883988105875 1415 82 C2 [4]
477 ⟨105  8   ⟩ 0.009345794393 0.148190714585 0.883241983098 1423 115 D1 [4]
478 ⟨ 86 10   1⟩ 0.011363704023 0.219552535348 0.883240835667 1410 98 D1 [4]
479 ⟨ 86 10   ⟩ 0.011363636364 0.219551228133 0.885083350098 1428 100 D2 [4]
480 ⟨ 79 11   ⟩ 0.012345679012 0.259908134361 0.884303207834 1429 92 C2 [4]
481 ⟨ 73 12   ⟩ 0.013333333333 0.303794795878 0.884284662208 1431 87 D1 [4]
482 ⟨106  8   ⟩ 0.009259259259 0.146818578338 0.884236393147 1439 118 D2 [4]
483 ⟨ 68 13   ⟩ 0.014285714286 0.350238007120 0.884173817721 1436 83 D1 [4]
484 ⟨ 87 10   ⟩ 0.011235955056 0.217084360401 0.884273651657 1442 99 D1 [4]
485 ⟨ 56 16   ⟩ 0.017241379310 0.512289877950 0.884137577266 1440 76 D2 [4]
486 ⟨ 80 11   ⟩ 0.012195121951 0.256738522967 0.884438010168 1447 93 D1 [4]
487 ⟨ 74 12   1⟩ 0.013157898728 0.299797586869 0.883535044475 1432 87 D1 [4]
488 ⟨ 74 12   ⟩ 0.013157894737 0.299797495932 0.885349016283 1453 90 D2 [4]
489 ⟨ 88 10   1⟩ 0.011111175797 0.214673561714 0.883487090029 1443 100 D1 [4]
490 ⟨ 88 10   ⟩ 0.011111111111 0.214672311952 0.885288658156 1461 102 D2 [4]
491 ⟨108  8   ⟩ 0.009090909091 0.144149149641 0.884369812511 1466 120 D2 [4]
492 ⟨ 81 11   ⟩ 0.012048192771 0.253645287750 0.884569564259 1465 94 C2 [4]
493 ⟨ 57 16   ⟩ 0.016949152542 0.503606998663 0.883488725642 1463 75 D1 [4]
494 ⟨ 75 12   ⟩ 0.012987012987 0.295904021959 0.884595046478 1470 89 D1 [4]
495 ⟨ 89 10   ⟩ 0.010989010989 0.212313275557 0.884494499249 1475 101 D1 [4]
496 ⟨ 61 15   ⟩ 0.015873015873 0.444139081167 0.883957653757 1473 78 C2 [4]
497 ⟨ 70 13   ⟩ 0.013888888889 0.340509173589 0.884529765291 1478 85 D1 [4]
498 ⟨ 82 11   ⟩ 0.011904761905 0.250625700991 0.884697986137 1483 95 D1 [4]
499 ⟨110  8   1⟩ 0.008928893469 0.141580164085 0.882761308647 1478 119 D1 [4]
500 ⟨110  8   ⟩ 0.008928571429 0.141575057683 0.884498466901 1493 122 D2 [4]





Updates

For updates look at the list below.

10-Dec-2011: First complete presentation from N=1 to N=500.

References

[1]   M. Ruda, The packing of circles in rectangles, (Hungarian. English summary), Magyar Tud. Akad. Mat. Fiz. Tud. Oszt., Közl 19 (1970), 73-87.
[2]   B.D. Lubachevsky, R. Graham, Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio, In B. Aronov, S. Basu, J. Pach and M. Sharir (eds.), Discrete and Computational Geometry: The Goldman-Pollack Festschrift, vol. 25 of Algorithms and Combinatorics, pages 633–650, Springer, 2003.
[3]   E.G. Birgin, J.M. Gentil, New and improved results for packing identical unitary radius circles within triangles, rectangles and strips, Comput. Oper. Res. 37 (2010), 1318-1327.
[4]   E. Specht, High density packings of equal circles in rectangles with variable aspect ratio, Comput. Oper. Res. 40 (2013), 58-69.


©  E. Specht     12-Sep-2012