Periodic sequence

In mathematics, a periodic sequence (sometimes called a cycle^{[ citation needed ]}) is a sequence for which the same terms are repeated over and over:

a₁, a₂, ..., a_p,  a₁, a₂, ..., a_p,  a₁, a₂, ..., a_p, ...^{[ citation needed ]}

The number p of repeated terms is called the period (period). ^[1]

Definition

A (purely) periodic sequence (with period p), or a p-periodic sequence, is a sequence a₁, a₂, a₃, ... satisfying

a_n+p = a_n

for all values of n. ^{[1] [2] [3] [4] [5]} If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.^{[ citation needed ]} The smallest p for which a periodic sequence is p-periodic is called its least period ^{[1] [6]} or exact period. ^{[6] [ verification needed ]}

Examples

Every constant function is 1-periodic. ^[4]

The sequence ${\displaystyle 1,2,1,2,1,2\dots }$ is periodic with least period 2. ^[2]

The sequence of digits in the decimal expansion of 1/7 is periodic with period 6:

${\displaystyle {\frac {1}{7}}=0.142857\,142857\,142857\,\ldots }$

More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below). ^{[7] [ verification needed ]}

The sequence of powers of −1 is periodic with period two:

${\displaystyle -1,1,-1,1,-1,1,\ldots }$

More generally, the sequence of powers of any root of unity is periodic. The same holds true for the powers of any element of finite order in a group.^{[ citation needed ]}

A periodic point for a function f : X → X is a point x whose orbit

${\displaystyle x,\,f(x),\,f(f(x)),\,f^{3}(x),\,f^{4}(x),\,\ldots }$

is a periodic sequence. Here, ${\displaystyle f^{n}(x)}$ means the n-fold composition of f applied to x. ^{[6] [ verification needed ]} Periodic points are important in the theory of dynamical systems. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point.^{[ citation needed ]}

Identities

Partial Sums

${\displaystyle \sum _{n=1}^{kp+m}a_{n}=k*\sum _{n=1}^{p}a_{n}+\sum _{n=1}^{m}a_{n}}$ Where k and m<p are natural numbers.^{[ citation needed ]}

Partial Products

${\displaystyle \prod _{n=1}^{kp+m}a_{n}=({\prod _{n=1}^{p}a_{n}})^{k}*\prod _{n=1}^{m}a_{n}}$ Where k and m<p are natural numbers.^{[ citation needed ]}

Periodic 0, 1 sequences

Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. Periodic zero and one sequences can be expressed as sums of trigonometric functions:

${\displaystyle \sum _{k=1}^{1}\cos(-\pi {\frac {n(k-1)}{1}})/1=1,1,1,1,1,1,1,1,1...}$

${\displaystyle \sum _{k=1}^{2}\cos(2\pi {\frac {n(k-1)}{2}})/2=0,1,0,1,0,1,0,1,0...}$

${\displaystyle \sum _{k=1}^{3}\cos(2\pi {\frac {n(k-1)}{3}})/3=0,0,1,0,0,1,0,0,1,0,0,1,0,0,1...}$

${\displaystyle ...}$

${\displaystyle \sum _{k=1}^{N}\cos(2\pi {\frac {n(k-1)}{N}})/N=0,0,0...,1{\text{ sequence with period }}N}$^{[ citation needed ][ clarification needed ]}

Generalizations

A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:

1 / 56 = 0 . 0 1 7  8 5 7 1 4 2  8 5 7 1 4 2  8 5 7 1 4 2  ...^{[ citation needed ]}

A sequence is ultimately periodic if it satisfies the condition ${\displaystyle a_{k+r}=a_{k}}$ for some r and sufficiently large k. ^[1]

A sequence is asymptotically periodic if its terms approach those of a periodic sequence. That is, the sequence x₁, x₂, x₃, ... is asymptotically periodic if there exists a periodic sequence a₁, a₂, a₃, ... for which

${\displaystyle \lim _{n\rightarrow \infty }x_{n}-a_{n}=0.}$ ^{[4] [8] [9] [ verification needed ]}

For example, the sequence

1 / 3,  2 / 3,  1 / 4,  3 / 4,  1 / 5,  4 / 5,  ...

is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....^{[ citation needed ]}

References

1. "Ultimately periodic sequence - Encyclopedia of Mathematics". encyclopediaofmath.org. 7 February 2011. Retrieved 13 August 2021.
2. Weisstein, Eric W. "Periodic Sequence". mathworld.wolfram.com. Retrieved 2021-08-13.
3. Bosma, Wieb. "Complexity of Periodic Sequences" (PDF). www.math.ru.nl. Retrieved 13 August 2021.
4. Janglajew, Klara; Schmeidel, Ewa (2012-11-14). "Periodicity of solutions of nonhomogeneous linear difference equations". Advances in Difference Equations. 2012 (1): 195. doi: 10.1186/1687-1847-2012-195 . ISSN   1687-1847. S2CID   122892501.
5. Menezes, Alfred J.; Oorschot, Paul C. van; Vanstone, Scott A. (2018-12-07). Handbook of Applied Cryptography. CRC Press. ISBN   978-0-429-88132-9.
6. Weisstein, Eric W. "Least Period". mathworld.wolfram.com. Retrieved 2021-08-13.
7. Hosch, William L. (1 June 2018). "Rational number". Encyclopedia Britannica. Retrieved 13 August 2021.
8. Cheng, SuiSun (2017-09-29). New Developments in Difference Equations and Applications: Proceedings of the Third International Conference on Difference Equations. Routledge. ISBN   978-1-351-42880-4.
9. Shlezinger, Nir; Todros, Koby (2019-01-01). "Performance analysis of LMS filters with non-Gaussian cyclostationary signals". Signal Processing. 154: 260–271. arXiv: 1708.00635 . doi:10.1016/j.sigpro.2018.08.008. ISSN   0165-1684. S2CID   53521677.