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# Index to Elementary Cellular Automata

Index to Elementary Cellular Automata

Sequences in the OEIS related to Elementary Cellular Automata are tabulated here.

## Introduction

There are 256 possible Elementary Cellular Automata. Wolfram's numbering scheme identifies these by rule numbers from 0 to 255. See "A New Kind of Science" by Stephen Wolfram, pages 51-60 or the Eric Weisstein's "World of Mathematics" links below. The sequences in the OEIS generally represent a one-dimensional cellular automata (CA) as a growing triangle whose rows represent successive generations or stages. These rows grow as 2n+1 where n=0 specifies the initial state of a single black (ON) cell. Black (ON) cells are translated as ones and white (OFF) cells as zeroes. For example, the first five rows of the CA generated by Rule 1 are:

```            1
0 0 0
0 0 1 0 0
1 1 0 0 0 1 1
0 0 0 0 1 0 0 0 0
```

Certain attributes of these CA can be interpreted as integer sequences. These are tabulated in the tables below. The most basic attribute is the binary interpretation of the entire generated triangle itself. For example, the sequence generated by Rule 1 begins: 1,0,0,0,0,0,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,0,0,0,0. Another sequence interprets each row as a binary number. The sequence generated by Rule 1 for this attribute begins: 1,0,100,1100011,10000. These binary numbers, when converted to decimal yields another sequence. For Rule 1, the sequence begins: 1,0,4,99,16. The middle column of the triangle can also be interpreted as a binary sequence of ones and zeroes. For Rule 1, the sequence begins: 1,0,1,0,1. This middle column can also be represented as a sequence of binary numbers as the triangle grows. For Rule 1, this sequence begins: 1,10,101,1010,10101. Again, the decimal equivalent produces another sequence. For Rule 1, this sequence begins: 1,2,5,10,21. Another sequence is the number of black (ON) cells at each generation of the CA. For Rule 1, this sequence begins: 1,0,1,4,1. Yet another sequence is obtained by counting the total number of black (ON) cells through n generations. For Rule 1, this sequence begins: 1,1,2,6,7. Counting the white (ON) cells instead of the black cells yields two more sequences. For Rule 1, these two sequences are: 0,3,4,3,8 and 0,3,7,10,18. Other sequences are defined by the left and right diagonals. The left diagonal for Rule 1 begins: 1,0,0,1,0,1,0,1. The right diagonal in this case is the same. But not all rules produce a symmetric triangle. Similar to the middle column sequences, a binary and decimal interpretation of the left and right diagonals produce yet other sequences. As it turns out, there are only six different sequences for each of the left and right diagonals. These sequences are tabulated in the tables below titled as "Left Diagonal Sequences" and "Right Diagonal Sequences".

Given the number of black cells in a row, Black(n), and the fact that there are 2n+1 cells in a row, certain sequences can be computed from others. Black(${\displaystyle \Sigma }$) is the partial sum of Black(n) and Black(n) is the first difference of Black(${\displaystyle \Sigma }$). Also, White(n) = 2n + 1 - Black(n). Similarly White(${\displaystyle \Sigma }$) is the partial sum of White(n) and White(n) is the first difference of White(${\displaystyle \Sigma }$). Also, White(n) = 2n + 1 - Black(n) and White(${\displaystyle \Sigma }$) = (n+1)^2 - Black(${\displaystyle \Sigma }$). Thus, if any of these four sequences are known, the other three can be calculated. And, if two or more CA generate an identical sequence for one of these, then all four must be identical for all of those CA.

Certain CA definitions (or rules), while different, generate identical histories. For example, rules 1 and 33 while having different specifications generate identical triangles. See the section below on "Equivalent Elementary Cellular Automata". Sequences generated by the middle column using various rules may also be identical. For example rules 60, 102, 110, 150 and 188 each show all black cells in the middle column. For another example, rules 57, 99, 131 and 145 all generate the same middle column. See "Equivalencies Based on the Middle Column " below for these sequences. Similarly, different rules can generate identical Black cell counts in each row. For example, rules 28, 70, 156 and 198 all generate the same Black(n) sequence. See "Equivalencies Based on Number of ON (Black) Cells in a Row" below for these sequences.

Often a well known sequence like A005408 (The Odd Integers) comes very close to the CA generated sequence. The first term, usually a zero, may be missing or substituted. Or an extra first term will be present. In these cases the well known sequence is used in the tables below and the sequence is indicated with an asterisk. For example, the binary sequence of the middle column as the triangle grows for Rule 1 is: 1,10,101,1010,10101,... . A056830(n) = 0,1,10,101,1010,10101,... . The CA sequence is A056830 ignoring the the first term of A056830.

Certain CA are mirror images of another. Some are near complements of each other, where black (ON) cells are swapped with white (OFF) cells. Still others are complements of the mirror image. Wolfram states that Rule 135 is the complement of Rule 30. This is not precisely true, as both start with a single black (ON) cell at stage 0. But if you compare Rule 135 starting at stage 1 ignoring the leftmost two cells, it matches the complement of Rule 30 starting at stage 0. This type of near complement is seen in rules 135 and 149 among others.

```          1                        1
1 1 0                    1 1 1                   0
1 1 0 0 0                1 1 0 0 1               0 0 0
1 1 0 0 1 1 0            1 1 0 1 1 1 1           0 0 1 1 0
1 1 0 0 1 0 0 0 0        1 1 0 0 1 0 0 0 1       0 0 1 0 0 0 0
1 1 0 0 1 1 0 1 1 1 0    1 1 0 1 1 1 1 0 1 1 1   0 0 1 1 0 1 1 1 0
Rule 135                 Rule 30          Truncated Rule 135
```

## Sequences in the OEIS Related to Elementary Cellular Automata

```Column headings in the table below:

Rule         Wolfram's Rule Number
Triangle     A-number of sequence listing the CA stages (iterations, generations) in a triangular table
Stages(B)    A-number of sequence listing each CA stage as a binary number
Stages(D)    A-number of sequence listing each CA stage as a decimal number
Middle       A-number of sequence listing the middle column of the CA stages in a binary representation
Middle(B)    A-number of sequence listing the middle column at each stage as a binary number
Middle(D)    A-number of sequence listing the middle column at each stage as a decimal number
Black(n)     A-number of sequence listing the number of black (ON) cells at each stage
Black(${\displaystyle \Sigma }$)     A-number of sequence listing the total number of black (ON) cells through stage n
White(n)     A-number of sequence listing the number of white (OFF) cells at each stage
White(${\displaystyle \Sigma }$)     A-number of sequence listing the total number of white (OFF) cells through stage n
Other        A-number of sequence listing other attributes of the CA
(an asterisk, "*", in an A-number indicates a near match)
```
Elementary Cellular Automata Sequences in the OEIS
Rule Triangle Stages(B) Stages(D) Middle Middle(B) Middle(D) Black(n) Black() White(n) White() Other
0,8,32,40,64,72,96, 104,128,136,160,168, 192,200,224,232 A000007 A000007 A000007 A000007 A011557 A000079 A000007 A000012 A005408* A005563
1,33 A265718 A265720 A265721 A059841 A056830* A000975* A265722 A128918 A265723 A265724
2,10,34,42,66,74,98, 106,130,138,162,170, 194,202,226,234 A010052 A098608 A000302 A000007 A011557 A000079 A000012 A000027 A005843 A002378
3,35 A263428 A266068 A266069 A266070 A266071 A081253* A266072 A247375 A266073 A266074
4,12,36,44,68,76, 100,108,132,140,164, 172,196,204,228,236 A005369 A011557 A000079 A000012 A002275* A168604 A000012 A000027 A005843 A002378
5 A266174 A266175 A266176 A000012 A002275* A168604 A266072 A247375 A266073 A266074
6,38,134,166 A266178 A266179 A266180 A019590 A003953 A003945 A000034 A032766* A042948 A035608
7 A266216 A266217 A266218 A135528 A266219 A081254 A266220 A266221 A266222 A266223
9 A266243 A266244 A266245 A266246 A266247 A266248 A266249 A266250 A266251 A266252
11,43 A266253 A266254 A266255 A266070 A266071 A081253* A266256 A266257 A266258 A266259
13 A266282 A266283 A266284 A000012 A002275* A168604 A266285 A053439 A266286 A266287
14,46,142,174 A266298 A266299 A164908 A019590 A003953 A003945 A040000 A005408 A004273 A000290
15 A266300 A266301 A266302 A135528 A266219 A081254 A266303 A131179* A265722 A266304
16,24,48,56,80,88, 112,120,144,152,176, 184,208,216,240,248 A065803 A000012 A000012 A000007 A011557 A000079 A000012 A000027 A005843 A002378
17,49 A260552 A260692 A266090 A266070 A266071 A081253* A266072 A247375 A266073 A266074
18,26,82,90, 146,154,210,218 A070886 A265172 A081253* A000007 A011557 A000079 A001316 A006046* A071042 A171378 A245191
19 A266155 A266323 A266324 A266070 A266071 A081253* A266220 A266221 A266222 A266223
20,52,148,180 A266326 A266327 A010684 A019590 A003953 A003945 A000034 A032766* A042948 A035608
21 A266377 A266379 A266380 A135528 A266219 A081254 A266220 A266221 A266222 A266223
22 A071029 A266381 A266382 A019590 A003953 A003945 A071044 A266383 A071043 A266384
23,31,55,63, 87,95,119,127 A266434 A266435 A266436 A135528 A266219 A081254 A266437 A266438 A266439 A266440
25 A266441 A266442 A266443 A266444 A266445 A266446 A266447 A266448 A266449 A266450
27 A266459 A266460 A266461 A266070 A266071 A081253* A266303 A131179* A265722 A266304
28,156 A266502 A266508 A001045* A000012 A002275* A168604 A080513 A024206* A032766 A006578
29 A266514 A266515 A266516 A000012 A002275* A168604 A266303 A131179* A265722 A266304
30 A070950 A245549 A110240 A051023 A261299 A092539 A070952 A110267 A070951 A265224
37 A266588 A266589 A266590 A266591 A266592 A052997 A266593 A266594 A266595 A266596
39 A266605 A266606 A266607 A135528 A266219 A081254 A266303 A131179* A265722 A266304
41 A266608 A266609 A266610 A266611 A266612 A266613 A266614 A266615 A266616 A266617
45 A266619 A266621 A266622 A266623 A266624 A266625 A266626 A266627 A266628 A266629
47 A266659 A266660 A266661 A135528 A266219 A081254 A266662 A266663 A266664 A266665
50,58,114,122,178, 179,186,242,250 A071028 A094028 A002450* A059841 A056830* A000975* A000027 A000217 A001477 A000217
51 A266666 A266667 A266668 A059841 A056830* A000975* A266303 A131179* A265722 A266304
53 A266669 A266670 A266671 A135528 A266219 A081254 A266303 A131179* A265722 A266304
54 A071030 A118109 A118108 A133872 A259661 A077854 A064455 A265225 A071045 A050187*
57 A266672 A266673 A266674 A079978 A033146 A033138 A266285 A053439 A266286 A266287
59 A266716 A266717 A266718 A266719 A266720 A266721 A266722 A266723 A266724 A266725
60 A075438 A006943 A001317 A000012 A002275* A168604 A001316 A006046* A071042 A171378 A047999
61 A266786 A266787 A266788 A266789 A266790 A266791 A266792 A266793 A266794 A266795
62 A071031 A266809 A266810 A011655* A033137 A033129* A071047 A266811 A071046 A266813
65 A266243 A266244 A266245 A266246 A266247 A266248 A266249 A266250 A266251 A266252
67 A266837 A266838 A266839 A266444 A266445 A266446 A266447 A266448 A266449 A266450
69 A266840 A266841 A266842 A000012 A002275* A168604 A266285 A053439 A266286 A266287
70,198 A266843 A266844 A266846 A000012 A002275* A168604 A080513 A024206* A032766 A006578
71 A266848 A266849 A266850 A000012 A002275* A168604 A266303 A131179* A265722 A266304
73 A262448 A265122 A265156 A059841 A056830* A000975* A265205 A265206 A265219 A265220
75 A266892 A266893 A266894 A266895 A266896 A266897 A266898 A266899 A266900 A266901
77 A059841 A266872 A266873 A000012 A002275* A168604 A109613 A000982* A052928* A007590*
78 A266974 A266975 A266976 A000012 A002275* A168604 A266977 A004116 A001651* A077043
79 A266978 A266979 A266980 A000012 A002275* A168604 A266981 A031878* A064455* A265225*
81,113 A266982 A266983 A266984 A266070 A266071 A081253* A266256 A266257 A266258 A266259
83 A267001 A267002 A267003 A266070 A266071 A081253* A266303 A131179* A265722 A266304
84,116,212,244 A267006 A010850* A010701* A019590 A003953 A003945 A040000 A005408 A004273 A000290
85 A267034 A267035 A267036 A135528 A266219 A081254 A266303 A131179* A265722 A266304
86 A071032 A265280 A265281 A051023 A261299 A092539 A070952 A110267 A070951 A265224
89 A267037 A267038 A267039 A266895 A266896 A266897 A266898 A266899 A266900 A266901
91 A267015 A267041 A267042 A267043 A267044 A267045 A267046 A267047 A267048 A267049
92 A267050 A267051 A267052 A000012 A002275* A168604 A266977 A004116 A001651* A077043
93 A267053 A267054 A267055 A000012 A002275* A168604 A266981 A031878* A064455* A265225*
94 A118102 A071033 A118101 A019590 A003953 A003945 A265283 A265284 A109613 A000982
97 A267056 A267057 A267058 A266611 A266612 A266613 A266614 A266615 A266616 A266617
99 A267126 A267127 A267128 A079978 A033146 A033138 A266285 A053439 A266286 A266287
101 A267129 A267130 A267131 A266623 A266624 A266625 A266626 A266627 A266628 A266629
102 A075439 A265319 A117998 A000012 A002275* A168604 A001316 A006046* A071042 A171378 A001317, A047999
103 A267136 A267138 A267139 A266789 A266790 A266791 A266792 A266793 A266794 A266795
105 A267145 A267146 A267147 A059841 A056830* A000975* A267148 A267149 A267150 A267151
107 A267152 A267153 A267154 A267155 A267156 A267157 A267158 A267159 A267160 A267161
109 A243560 A267206 A267207 A267208 A267209 A267210 A267211 A267212 A267213 A267214
110 A075437 A265320 A117999 A000012 A002275* A168604 A071049 A265321 A265322 A265323 A070887, A006978
111 A267253 A267254 A267255 A267256 A267257 A267258 A267259 A267260 A267261 A267262
115 A267269 A267270 A267271 A266719 A266720 A266721 A266722 A266723 A266724 A266725
117 A267272 A267273 A267274 A135528 A266219 A081254 A266662 A266663 A266664 A266665
118 A071034 A267275 A267276 A011655 A033137 A033129 A071047 A266811 A266812 A266813
121 A267292 A267293 A267294 A267155 A267156 A267157 A267158 A267159 A267160 A267161
123 A267349 A267350 A267351 A059841 A056830* A000975* A267352 A267353 A267354 A131179
124 A267355 A267356 A267357 A000012 A002275* A168604 A071049 A265321 A265322 A265323
125 A267358 A267359 A267360 A267256 A267257 A267258 A267259 A267260 A267261 A267262
126 A071035 A267364 A267365 A036987 A267366 A267367 A071051 A267368 A071050 A267369
129,161 A267417 A267440 A267441 A267442 A267443 A267444 A267445 A267446 A267447 A267448
131 A267418 A267449 A267450 A079978 A033146 A033138 A267451 A267452 A267453 A267454
133 A267423 A267456 A267457 A000012 A002275* A168604 A267458 A267459 A267460 A267461
135 A265695 A265696 A265697 A265698 A265699 A265700 A265701 A265702 A265703 A265704 A226463
137 A267463 A267511 A267512 A267513 A267514 A267515 A267516 A267517 A267518 A267519
139,171 A267520 A267523 A156760* A179184 A267524 A054135* A005408* A002522 A007395* A005843
141 A267525 A267526 A267527 A000012 A002275* A168604 A267528 A267529 A2675230 A2675231
143 A267533 A267535 A267536 A267537 A267538 A267539 A004280 A010000 A130130 A004273
145 A262805 A262859 A262860 A079978 A033146 A033138 A267451 A267452 A267453 A267454
147 A262808 A262861 A262862 A133872* A262863 A262864 A266981 A031878* A064455* A265225*
149 A265246 A265715 A265717 A265698 A265699 A265700 A265701 A265702 A070952 A110267 A226464
150 A071036 A118110 A038184 A000012 A002275* A168604 A071053 A134659 A071052 A265223
Elementary Cellular Automata Sequences in the OEIS (continued)
Rule Triangle Stages(B) Stages(D) Middle Middle(B) Middle(D) Black(n) Black() White(n) White() Other
151,159,183,191,215, 222,223,247,254,255 A000012 A100706 A083420 A000012 A002275* A168604 A005408 A000290* A000004 A000004
153 A262855 A262865 A262866 A000007 A011557 A000079 A071042* A262867 A001316* A074330*
155 A263243 A263244 A263245 A179184 A267524 A054135* A053438 A263511 A134451 A032766
157 A263804 A263805 A263806 A000012 A002275* A168604 A007494* A263807 A008619* A024206
158 A071037 A265379 A118171 A166486 A265380 A265381 A071054 A265382 A029578 A211538
163 A263919 A266752 A266753 A266070 A266071 A081253* A028310 A000124 A020725* A000096
165 A266754 A267246 A267247 A000012 A002275* A168604 A071042* A262867 A001316* A074330*
167 A267576 A267577 A267578 A267579 A267580 A267581 A267582 A267583 A001316* A006046
169 A264442 A267585 A267586 A267587 A267588 A267589 A267590 A267591 A267592 A267593
173 A267594 A267595 A267596 A000012 A002275* A168604 A103517* A103505* A054977* A000027*
175 A265186 A262779 A198694* A266678 A266680 A267404 A004277 A002061* A057427 A001477
177 A267598 A267599 A083584* A266070 A266071 A081253* A028310 A000124 A020725* A000096
181 A267605 A267606 A267607 A267579 A267580 A267581 A267582 A267583 A001316* A006046
182 A071038 A267608 A267609 A000012 A002275* A168604 A071042* A171378* A071055 A267610
185 A267612 A267613 A267614 A179184 A267524 A054135* A005408* A002522 A007395* A005843
187 A267621 A267622 A140529* A060576 A267623 A083329 A004277 A002061* A057427 A001477
188 A118174 A265427 A118173 A000012 A002275* A168604 A265428 A265429 A265430 A265431
189 A267635 A099814* A103454 A000012 A002275* A168604 A004277 A002061* A057427 A001477
190 A118111 A265688 A037576 A166486 A265380 A265381 A032766 A006578 A004526 A002620
193 A267636 A267645 A267646 A267513 A267514 A267515 A267516 A267517 A267518 A267519
195 A267673 A267674 A267675 A000007 A011557 A000079 A071042* A262867 A001316* A074330*
197 A267676 A267677 A267678 A000012 A002275* A168604 A267528 A267529 A2675230 A2675231
199 A267687 A267688 A267689 A000012 A002275* A168604 A007494* A263807 A008619* A024206
201 A267679 A267680 A267681 A059841 A056830* A000975* A014601 A267682 A176059* A047218
203 A267683 A267684 A267685 A000007 A011557 A000079 A103517* A103505* A054977* A000027*
205 A267704 A267705 A188530* A000012 A002275* A168604 A005408* A002522 A007395* A005843
206,238 A267708 A109241 A171476 A000012 A002275* A168604 A000027 A000217 A001477 A000217
207 A267773 A267775 A267774 A000012 A002275* A168604 A004277 A002061* A057427 A001477
209,241 A267776 A267777 A141722* A179184 A267524 A054135* A005408* A002522 A007395* A005843
211 A267778 A267779 A267780 A179184 A267524 A054135* A053438 A263511 A134451 A032766
213 A267800 A267801 A267802 A267537 A267538 A267539 A004280 A010000 A130130 A004273
214 A071040 A267804 A267805 A166486 A265380 A265381 A071054 A265382 A029578 A211538
217 A267810 A267811 A267812 A000007 A011557 A000079 A103517* A103505* A054977* A000027*
219 A267813 A138148* A129868* A000007 A011557 A000079 A004277 A002061* A057427 A001477
220,252 A118175 A002275* A000225* A000012 A002275* A168604 A000027 A000217 A000027 A000217 A219238
221 A267814 A267815 A267816 A000012 A002275* A168604 A004277 A002061* A057427 A001477
225 A267841 A267842 A267843 A267587 A267588 A267589 A267590 A267591 A267592 A267593
227 A267845 A267846 A267847 A179184 A267524 A054135* A005408* A002522 A007395* A005843
229 A267848 A267850 A267851 A000012 A002275* A168604 A103517* A103505* A054977* A000027*
230 A267853 A267854 A267855 A000012 A002275* A168604 A265428 A265429 A265430 A265431
231 A267866 A267867 A002446* A000012 A002275* A168604 A004277 A002061* A057427 A001477
233 A267868 A267876 A267877 A267878 A267879 A267880 A267881 A267882 A267883 A267884
235 A267869 A267885 A267886 A179184 A267524 A054135* A267873 A267874 A033322* A157532*
237 A267870 A267887 A267888 A000012 A002275* A168604 A267872 A008865* A000038 A007395*
239 A267871 A267889 A267890 A000012 A002275* A168604 A140139 A005563* A063524 A057427
243 A267919 A267920 A267921 A060576 A267623 A083329 A004277 A002061* A057427 A001477
245 A267922 A267923 A267924 A266678 A266680 A267404 A004277 A002061* A057427 A001477
246 A071041 A267925 A267926 A166486 A265380 A265381 A032766 A006578 A004526 A002620
249 A267927 A267934 A267935 A179184 A267524 A054135* A267873 A267874 A033322* A157532*
251 A267936 A267937 A267938 A060576 A267623 A083329 A140139 A005563* A063524 A057427
253 A060576 A267940 A267941 A000012 A002275* A168604 A140139 A005563* A063524 A057427

Left Diagonal Sequences

```Column headings in the table below:

Rule         Wolfram's Rule Number
L_Diag       A-number of sequence listing the left diagonal in a binary representation
L_Diag(B)    A-number of sequence listing the left diagonal at each stage as a binary number
L_Diag(D)    A-number of sequence listing the left diagonal at each stage as a decimal number
(an asterisk, "*", in an A-number indicates a near match)
```
Elementary Cellular Automata Sequences in the OEIS (Left Diagonal)
Rule L_Diag L_Diag(B) L_Diag(D)
0,4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100,104,108, 112,116,120,124,128,129,132,133,136,137,140,141,144,145,148,149,152,153,156,157,160,161,164,165, 168,169,172,173,176,177,180,181,184,185,188,189,192,196,200,204,208,212,216,220,224,228,232,236, 240,244,248,252 A000007 A011557 A000079
1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61 A266070 A266071 A081253*
2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74,78,82,86,90,94,98,102,106,110,114,118,122, 126,130,131,134,135,138,139,142,143,146,147,150,151,154,155,158,159,162,163,166,167,170,171,174, 175,178,179,182,183,186,187,190,191,194,195,198,199,202,203,206,207,210,211,214,215,218,219,222, 223,226,227,230,231,234,235,238,239,242,243,246,247,250,251,254,255 A000012 A002275* A168604
3,7,11,15,19,23,27,31,35,39,43,47,51,55,59,63,67,71,75,79,83,87, 91,95,99,103,107,111,115,119,123, 127 A135528 A266219 A081254
65,69,73,77,81,85,89,93,97,101,105,109,113,117,121,125 A059841 A056830* A000975*
193,197,201,205,209,213,217,221,225,229,233,237,241,245,249,253 A060576 A267623 A083329

Right Diagonal Sequences

```Column headings in the table below:

Rule         Wolfram's Rule Number
R_Diag       A-number of sequence listing the right diagonal in a binary representation
R_Diag(B)    A-number of sequence listing the right diagonal at each stage as a binary number
R_Diag(D)    A-number of sequence listing the right diagonal at each stage as a decimal number
(an asterisk, "*", in an A-number indicates a near match)
```
Elementary Cellular Automata Sequences in the OEIS (Right Diagonal)
Rule R_Diag R_Diag(B) R_Diag(D)
0,2,4,6,8,10,12,14,32,34,36,38,40,42,44,46,64,66,68,70,72,74,76,78,96,98,100,102,104,106, 108,110,128,129,130,131,132,133,134,135,136,138,140,142,160,161,162,163,164,165,166,167,168, 170,172,174,192,193,194,195,196,197,198,199,200,202,204,206,224,225,226,227,228,229,230,231, 232,234,236,238 A000007 A011557 A000079
1,3,5,7,33,35,37,39,65,67,69,71,97,99,101,103 A266070 A266071 A081253*
9,11,13,15,41,43,45,47,73,75,77,79,105,107,109,111 A059841 A056830* A000975*
16,18,20,22,24,26,28,30,48,50,52,54,56,58,60,62,80,82,84,86,88,90,92,94,112,114,116,118,120, 122,124,126,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,176,177,178,179, 180,181,182,183,184,185,186,187,188,189,190,191,208,209,210,211,212,213,214,215,216,217,218, 219,220,221,222,223,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255 A000012 A002275* A168604
17,19,21,23,25,27,29,31,49,51,53,55,57,59,61,63,81,83,85,87,89,91,93,95,113,115,117,119,121,123,125,127 A135528 A266219 A081254
137,139,141,143,169,171,173,175,201,203,205,207,233,235,237,239 A060576 A267623 A083329

## Mathematica Program To Generate the Sequences Above

rule=73; rows=20;

catri=Table[Take[ca[ [k] ],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *)

Print["CA Evolution Diagram"];

For[i=1, i≤rows, i++, str=StringTake[" ",rows-i]; str = str <> str; (* Create padding for particular row *)

For[j=1, j<=Length[catri[ [i] ]], j++, str = str <> If[catri[ [i] ][ [j] ] == 0, "0 ", "1 "]];

Print[str]];

Print["Triangle Representation of CA"]; Flatten[catri]

Print["Binary Representation of Rows"]; Table[FromDigits[catri[ [k] ]],{k,1,rows}]

Print["Decimal Representation of Rows"]; Table[FromDigits[catri[ [k] ],2],{k,1,rows}]

Print["Middle Column"]; mc=Table[catri[ [k] ][ [k] ],{k,1,rows}] (* Keep only middle cell from each row *)

Print["Binary Representation of Middle Column"]; Table[FromDigits[Take[mc,k]],{k,1,rows}]

Print["Decimal Representation of Middle Column"]; Table[FromDigits[Take[mc,k],2],{k,1,rows}]

Print["Number of Black cells in stage n"]; nbc=Table[Total[catri[ [k] ]],{k,1,rows}]

Print["Number of Black cells through stage n"]; Table[Total[Take[nbc,k]],{k,1,rows}]

Print["Number of White cells in stage n"]; nwc=Table[Length[catri[ [k] ]]-nbc[ [k] ],{k,1,rows}]

Print["Number of White cells through stage n"]; Table[Total[Take[nwc,k]],{k,1,rows}]

Print["Left Diagonal"]; ld=Table[catri[ [k] ][ [1] ], {k,1,rows}] (* Keep only first cell from each row *)

Print["Binary Representation of Left Diagonal"]; Table[FromDigits[Take[ld,k]], {k,1,rows}]

Print["Decimal Representation of Left Diagonal"]; Table[FromDigits[Take[ld,k],2], {k,1,rows}]

Print["Right Diagonal"]; rd=Table[catri[k][2*(k-1)+1], {k,1,rows}] (* Keep only last cell from each row *)

Print["Binary Representation of Right Diagonal"]; Table[FromDigits[Take[rd,k]], {k,1,rows}]

Print["Decimal Representation of Right Diagonal"]; Table[FromDigits[Take[rd,k],2], {k,1,rows}]

CA Evolution Diagram

```                                      1
0 0 0
1 0 1 0 1
0 0 0 0 0 0 0
1 0 1 1 1 1 1 0 1
0 0 0 1 0 0 0 1 0 0 0
1 0 1 0 0 0 1 0 0 0 1 0 1
0 0 0 0 0 1 0 0 0 1 0 0 0 0 0
1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 0 1
0 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0
1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1
0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0
1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1
0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0
1 0 1 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 0 0 0 0 0 1 0 1
0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 0 0
1 0 1 1 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 1 0 1
0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0
1 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 0 0 0 0 0 1 0 1
0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0
```

Triangle Representation of CA

{1,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0, 1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,1,1,1,0,0,0,1,0,0,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0, 1,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,1,1,1,1,1,0,0,0,1,1,1,1, 1,0,0,0,0,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0, 0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0, 1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,1,1,0,0,0,0,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0, 0,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0,1,0,1,0, 0,0,1,0,1,0,0,0,0,0,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,1, 1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1,1,0,0,0,0,0}

Binary Representation of Rows

{1,0,10101,0,101111101,10001000,1010001000101,1000100000,10111000100011101,1010100010101000, 101000000010000000101,111110001111100000,1011101000101010001011101,101000100000001000101000, 10100000100011111000100000101,11100010100010100011100000,101110101010000010000010101011101, 10100000001110001110000000101000,1010000011111010101010101111100000101, 1110100010000000000010001011100000}

Decimal Representation of Rows

{1,0,21,0,381,136,5189,544,94493,43176,1311749,254944,24400989,10617384,336720133,59386080, 6262162781,2688081960,86425034501,15602819808}

Middle Column

{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}

Binary Representation of Middle Column

{1,10,101,1010,10101,101010,1010101,10101010,101010101,1010101010,10101010101,101010101010, 1010101010101,10101010101010,101010101010101,1010101010101010,10101010101010101, 101010101010101010,1010101010101010101,10101010101010101010}

Decimal Representation of Middle Column

{1,2,5,10,21,42,85,170,341,682,1365,2730,5461,10922,21845,43690,87381,174762,349525,699050}

Number of Black cells in stage n

{1,0,3,0,7,2,5,2,9,6,5,10,13,6,11,10,15,10,19,10}

Number of Black cells through stage n

{1,1,4,4,11,13,18,20,29,35,40,50,63,69,80,90,105,115,134,144}

Number of White cells in stage n

{0,3,2,7,2,9,8,13,8,13,16,13,12,21,18,21,18,25,18,29}

Number of White cells through stage n

{0,3,5,12,14,23,31,44,52,65,81,94,106,127,145,166,184,209,227,256}

Left Diagonal

{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}

Binary Representation of Left Diagonal

{1,10,101,1010,10101,101010,1010101,10101010,101010101,1010101010,10101010101, 101010101010,1010101010101,10101010101010,101010101010101,1010101010101010, 10101010101010101,101010101010101010,1010101010101010101,10101010101010101010}

Decimal Representation of Left Diagonal

{1,2,5,10,21,42,85,170,341,682,1365,2730,5461,10922,21845,43690,87381,174762, 349525,699050}

Right Diagonal

{1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0}

Binary Representation of Right Diagonal

{1,10,101,1010,10101,101010,1010101,10101010,101010101,1010101010,10101010101, 101010101010,1010101010101,10101010101010,101010101010101,1010101010101010, 10101010101010101,101010101010101010,1010101010101010101,10101010101010101010}

Decimal Representation of Right Diagonal

{1,2,5,10,21,42,85,170,341,682,1365,2730,5461,10922,21845,43690,87381,174762, 349525,699050}

## Equivalent Elementary Cellular Automata

Rules in each of the lists below generate identical Cellular Automata histories:

{0,8,32,40,64,72,96,104,128,136,160,168,192,200,224,232}

{1,33}

{2,10,34,42,66,74,98,106,130,138,162,170,194,202,226,234}

{3,35}

{4,12,36,44,68,76,100,108,132,140,164,172,196,204,228,236}

{6,38,134,166}

{11,43}

{14,46,142,174}

{16,24,48,56,80,88,112,120,144,152,176,184,208,216,240,248}

{17,49}

{18,26,82,90,146,154,210,218}

{20,52,148,180}

{23,31,55,63,87,95,119,127}

{28,156}

{50,58,114,122,178,179,186,242,250}

{70,198}

{81,113}

{84,116,212,244}

{129,161}

{139,171}

{151,159,183,191,215,222,223,247,254,255}

{206,238}

{209,241}

{220,252}

## Equivalencies Based on Number of ON (Black) Cells in a Row

Rules in each of the lists below generate identical number of ON (Black) cells in a row. And as a consequence, the four related sequences are identical.

{2,4,10,12,16,24,34,36,42,44,48,56,66,68,74,76,80,88,98,100,106,108,112,120,130,132, 138,140,144,152,162,164,170,172,176,184,194,196,202,204,208,216,226,228,234,236,240,248}

{3,5,17,35,49}

{6,20,38,52,134,148,166,180}

{7,19,21}

{9,65}

{11,43,81,113}

{13,57,69,99}

{14,46,84,116,142,174,212,244}

{15,27,29,39,51,53,71,83,85}

{18,26,60,82,90,102,146,154,210,218}

{22}

{23,31,55,63,87,95,119,127}

{25,67}

{28,70,156,198}

{30,86}

{37}

{41,97}

{45,101}

{47,117}

{50,58,114,122,178,179,186,206,220,238,242,250,252}

{54}

{59,115}

{61,103}

{62,118}

{73}

{75,89}

{77}

{78,92}

{79,93,147}

{91}

{94}

{105}

{107,121}

{109}

{110,124}

{111,125}

{123}

{126}

{129,161}

{131,145}

{133}

{135,149}

{137,193}

{139,171,185,205,209,227,241}

{141,197}

{143,213}

{150}

{151,159,183,191,215,222,223,247,254,255}

{153,165,195}

{155,211}

{157,199}

{158,214}

{163,177}

{167,181}

{169,225}

{173,203,217,229}

{175,187,189,207,219,221,231,243,245}

{182}

{188,230}

{190,246}

{201}

{233}

{235,249}

{237}

{239,251,253}

## Equivalencies Based on the Middle Column

Rules in each of the lists below generate identical Middle Columns. And as a consequence, the three related sequences are identical.

{0,2,8,10,16,18,24,26,32,34,40,42,48,56,64,66,72,74,80,82,88,90,96,98,104,106, 112,120,128,130,136,138,144,146,152,153,154,160,162,168,170,176,184,192,194, 195,200,202,203,208,210,216,217,218,219,224,226,232,234,240,248}

{1,33,50,51,58,73,105,114,122,123,178,179,186,201,242,250}

{3,11,17,19,27,35,43,49,81,83,113,163,177}

{4,5,12,13,28,29,36,44,60,68,69,70,71,76,77,78,79,92,93,100,102,108,110,124, 132,133,140,141,150,151,156,157,159,164,165,172,173,182,183,188,189,191,196, 197,198,199,204,205,206,207,215,220,221,222,223,228,229,230,231,236,237,238, 239,247,252,253,254,255}

{6,14,20,22,38,46,52,84,94,116,134,142,148,166,174,180,212,244}

{7,15,21,23,31,39,47,53,55,63,85,87,95,117,119,127}

{9,65}

{25,67}

{30,86}

{37}

{41,97}

{45,101}

{54}

{57,99,131,145}

{59,115}

{61,103}

{62,118}

{75,89}

{91}

{107,121}

{109}

{111,125}

{126}

{129,161}

{135,149}

{137,193}

{139,155,171,185,209,211,227,235,241,249}

{143,213}

{147}

{158,190,214,246}

{167,181}

{169,225}

{175,245}

{187,243,251}

{233}