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Index to OEIS: Section Periodic

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Index to OEIS: Section Periodic

This is a special index page, which contains the "periodic sequences" subsection of the Index_to_OEIS:_Section_Per

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Index to (eventually) periodic sequences:

Periodic (or eventually periodic) sequences are linear recurrences with constant coefficients, namely a(n) = a(n-P), which corresponds to signature (0,...,0,1) (order P), but they can also satisfy a linear recurrence relation of order strictly lower than P, see examples below.

Period of length less than 10

(Entries are sorted according to the period length, the order of the linear recurrence they satisfy, and the signature of that recurrence relation.)

1-periodic = (eventually) constant sequences
see Index to (eventually) constant sequences
2-periodic sequences
These are linear recurrences, either of order 1, signature (-1) or of order 2, signature (0,1):
signature (-1), a.k.a. anti-periodic, a(n) = -a(n-1): A033999, A062157, A105812, A117154, A165326, A280560.
signature (0,1): A000034, A000035, A008899, A010170, A010673, A010674, A010675, A010676, A010677, A010678, A010679, A010680, A010681, A010684, A010685, A010686, A010688, A010689, A010690, A010694, A010695, A010696, A010697, A010698, A010699, A010700, A010702, A010703, A010704, A010705, A010706, A010707, A010708, A010710, A010711, A010712, A010713, A010714, A010715, A010717, A010718, A010719, A010720, A010721, A010723, A010724, A010725, A010726, A010728, A010729, A010730, A010732, A010733, A010735, A021444, A040001, A040003, A040005, A040007, A040008, A040011, A040013, A040015, A040019, A040021, A040024, A040029, A040031, A040032, A040033, A040035, A040041, A040043, A040048, A040055, A040057, A040059, A040063, A040071, A040073, A040074, A040077, A040080, A040091, A040093, A040094, A040099, A040109, A040111, A040120, A040131, A040133, A040134, A040135, A040137, A040139, A040143, A040155, A040157, A040168, A040181, A040183, A040185, A040188, A040195, A040209, A040211, A040212, A040214, A040215, A040219, A040224, A040239, A040241, A040243, A040247, A040255, A040271, A040273, A040288, A040305, A040307, A040308, A040309, A040311, A040314, A040317, A040323, A040341, A040343, A040360, A040379, A040381, A040383, A040384, A040387, A040389, A040399, A040419, A040421, A040422, A040425, A040426, A040433, A040440, A040461, A040463, A040465, A040472, A040483, A040505, A040507, A040528, A040551, A040553, A040554, A040555, A040557, A040559, A040563, A040567, A040575, A040599, A040601, A040604, A040609, A040624, A040649, A040651, A040653, A040662, A040675, A040701, A040703, A040704, A040707, A040710, A040719, A040728, A040755, A040757, A040759, A040762, A040763, A040769, A040783, A040811, A040813, A040840, A040869, A040871, A040872, A040873, A040874, A040875, A040879, A040881, A040884, A040889, A040899, A040929, A040931, A040960, A049071, A059841, A066711, A103947, A105397, A105398, A115634, A118536, A123138, A126664, A129000, A153284, A163522, A165734, A166024, A168309, A168330, A168361, A168428, A176040, A176260, A176415, A203777, A215036, A216125, A226294, A266327, A266444, A267319, A272664, A280193, A289203
3-periodic sequences
Linear recurrences of order 2, signature (-1,-1): A049347, A057078, A061347, A099837, A099838, A102283, A106510, A122434, A131713, A132677, A163804, A167373
order 3, signature (0,0,1): A008876, A008877, A008878, A008879, A008880, A008882, A008883, A008884, A010133, A010872, A010882, A011655, A021115, A021337, A021559, A021670, A033478, A033480, A040118, A040252, A040350, A040436, A040670, A040954, A052901, A069705, A070403, A070421, A073636, A078412, A079105, A079978, A080425, A082204, A100063, A100402, A101825, A105395, A109007, A110044, A118517, A118619, A118635, A130196, A130784, A130793, A130794, A131294, A131534, A131561, A131598, A131756, A136619, A141571, A144437, A146325, A153727, A164359, A164360, A165942, A166925, A167176, A168399, A169609, A173259, A173857, A175833, A177056, A177702, A204418, A209878, A214395, A224317, A239141, A244550, A244893, A257075, A271378
4-periodic sequences
Linear recurrences of order 2, signature (0,-1): ...
order 3, signature (-1,-1,-1):
order 3, signature (1,-1,1):
order 4, signature (0,0,0,1): ...
5-periodic sequences
Linear recurrences of order 4, signature (-1,-1,-1,-1)
order 5, signature (0,0,0,0,1).

Period of length ≥ 10

(Entries are sorted according to the period length, the order of the linear recurrence they satisfy, and the signature of that recurrence relation.)

12-periodic sequences

A089746 (repeat(4,4,1,2,1,1,2,2,3,3,3,3): number of syllables in English name of n-th month).

Period length ≥ 100

(Entries are sorted according to the period length, the order of the linear recurrence they satisfy, and the signature of that recurrence relation.)

247-periodic sequences
LR order 216, signature (1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,...): A014256 (1/Φ(247), cf Index to inverse of cyclotomic polynomials),
253-periodic sequences
LR order 220, signature (1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,-1,1,0,...): A014262 (1/Φ253, cf Index to inverse of cyclotomic polynomials),
259-periodic sequences
LR order 216, signature (1,1,1,1,1,1,1,0,...(total of 30 '0's)...,-1,-1,-1,-1,-1,-1,-1,0,0,0,...): A014268 (1/Φ(259), cf Index to inverse of cyclotomic polynomials),
399-periodic sequences
LR order 216, signature (1,-1,0,1,-1,0,1,0,-1,1,0,-1,1,0,0,0,0,0,0,1,-1,0,1,-1,0,1,0,-1,1,0,-1,1,0,0...): A014408 (1/Φ(399), cf Index to inverse of cyclotomic polynomials),
494-periodic sequences
LR order 216, signature (1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,0,0,0,0,0,0,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,0,0,0...): A014503 (1/Φ(494), cf Index to inverse of cyclotomic polynomials),
506-periodic sequences
LR order 220, signature (-1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,...): A014515 (1/Φ506, cf Index to inverse of cyclotomic polynomials),
518-periodic sequences
LR order 216, signature (1,-1,1,-1,1,-1,1,0,...(total of 30 '0's)...,1,-1,1,-1,1,-1,1,0,0,0,...): A014527 (1/Φ(518), cf Index to inverse of cyclotomic polynomials),
532-periodic sequences
LR order 216, signature (1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,...(total of 25 "0"s)...,1,0,-1,0,...): A014541 (1/Φ(532), cf Index to inverse of cyclotomic polynomials),
798-periodic sequences
LR order 216, signature (1,1,0,-1,-1,0,1,0,-1,-1,0,1,1,0,0,0,0,0,0,-1,-1,0,1,1,0,-1,0,1,1,0,-1,-1,0,...): A014807 (1/Φ(798), cf Index to inverse of cyclotomic polynomials),

Period of length ≥ 1000

1111-periodic sequences
A015120

Period of length ≥ 10^4

(Entries are sorted according to the period length, the order of the linear recurrence they satisfy, and the signature of that recurrence relation.)

16383-periodic sequences
order P = 16383, signature (0,...,0,1): A011727,
32767-periodic sequences
order P = 32767, signature (0,...,0,1): A011728,
65535-periodic sequences
order P = 65535, signature (0,...,0,1): A011729,
131071-periodic sequences
order 131071, signature (0,...,0,1): A011730,

Period of length ≥ 10100

LR order 23# · 277# ≈ 1.956858·10^121, signature (0,...,0,1): A000790 (primary pretenders: composite c s.th. n^c = n (mod c)).

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