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Triangle read by rows in which row n lists the possible eventual periods of cellular automaton rule 110 in a cyclic universe of width n.
3

%I #13 Jul 21 2023 23:31:46

%S 1,1,1,1,2,1,1,9,1,14,1,2,8,16,1,3,7,1,5,15,25,1,7,110,1,2,9,18,1,351,

%T 1,7,12,14,21,91,1,295,1,2,8,16,24,32,1,7,119,578,1,3,7,9,27,81,1,190,

%U 285,1,2,5,7,15,25,30,50,200,240,1,14,21,189,315,630

%N Triangle read by rows in which row n lists the possible eventual periods of cellular automaton rule 110 in a cyclic universe of width n.

%C Frequency of occurrence for the first 73 terms (semicolons separate rows): 2; 4; 8; 4, 12; 32; 10, 54; 9, 119; 20, 12, 8, 216; 17, 18, 477; 134, 220, 130, 540; 35, 495, 1518; 34, 12, 3426, 624; 54, 8138; 67, 8442, 644, 371, 168, 6692; 113, 32655; 116, 12, 8, 680, 1920, 62800; 138, 93330, 15895, 21709; 181, 36, 2349, 57024, 198594, 3960; 249, 28481, 495558; 534, 12, 3040, 49700, 270, 3300, 614140, 30660, 342380, 4540; 414, 455, 42, 7938, 331590, 1756713. Note that the sum of the frequencies of row n is 2^n.

%e Triangle begins:

%e 1

%e 1

%e 1

%e 1 2

%e 1

%e 1 9

%e 1 14

%e 1 2 8 16

%e 1 3 7

%e 1 5 15 25

%e 1 7 110

%e 1 2 9 18

%e 1 351

%e 1 7 12 14 21 91

%e 1 295

%e 1 2 8 16 24 32

%e 1 7 119 578

%e 1 3 7 9 27 81

%e 1 190 285

%e 1 2 5 7 15 25 30 50 200 240

%e 1 14 21 189 315 630

%Y Cf. A332718 (row lengths), A204371 (final terms for each row).

%K nonn,tabf,hard,more

%O 1,5

%A _Hans Havermann_, Jun 08 2020