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Eventually periodic sequences
This is the latest Eventually periodic sequences (or ultimately periodic sequences) are sequences for which there are some integers M and N such that, for all n > M, a(n) = a(n - N). The number N is called the period of the sequence, and the first M - N terms are called the preperiodic part of the sequence.
This definition includes periodic sequences and finite sequences as special cases.
Linear recurrences
All eventually periodic sequences with period N are linear recurrence relations of order at most N. The following tables give the possible combinations of period and order with their associated signatures.
| Order | Possible periods (with associated signatures up to 5) |
|---|---|
| 0 | 1: () |
| 1 | 1: (1); 2: (-1) |
| 2 | 2: (0,1); 3: (-1,-1); 4: (0,-1); 6: (1,-1) |
| 3 | 3: (0,0,1); 4: (1,-1,1), (-1,-1,-1); 6: (2,-2,1), (0,0,-1), (-2,-2,-1) |
| 4 | 4: (0,0,0,1); 5: (-1,-1,-1,-1); 6: (-1,0,1,1), (1,0,-1,1), (0,-1,0,-1); 8: (0,0,0,-1); 10: (1,-1,1,-1); 12: (1,-2,1,-1), (0,1,0,-1), (-1,-2,-1,-1) |
| 5 | 5: (0,0,0,0,1); 6: (1,-1,1,-1,1), (-1,-1,-1,-1,-1); 8: (1,0,0,-1,1), (-1,0,0,-1,-1); 10: (2,-2,2,-2,1), (0,0,0,0,-1), (-2,-2,-2,-2,-1); 12: (0,-1,1,0,1), (1,1,-1,-1,1), (2,-3,3,-2,1), (0,-1,-1,0,-1), (-1,1,1,-1,-1), (-2,-3,-3,-2,-1) |
| 6 | 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24, 30 |
| 7 | 7, 8, 9, 12, 14, 15, 18, 20, 24, 30 |
| 8 | 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 36, 40, 42, 60 |
| 9 | 9, 10, 12, 15, 16, 18, 20, 21, 24, 28, 30, 36, 40, 42, 60 |
| 10 | 10, 11, 12, 15, 16, 18, 20, 22, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 70, 72, 84, 90, 120 |
Charles R Greathouse IV, Eventually periodic sequences. — From the On-Line Encyclopedia of Integer Sequences® (OEIS®) wiki. (Available at https://oeis.org/wiki/Eventually_periodic_sequences)
