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Periodic sequence

A periodic sequence is a sequence of values, typically numbers or symbols, that repeats its pattern indefinitely after a fixed number of terms, known as the . Formally, a \{a_n\}_{n=1}^\infty is periodic with p (where p is a positive ) if a_{n+p} = a_n for all n \geq 1. The smallest such p is called the least period or fundamental of the . For instance, the $1, 2, 1, 2, 1, 2, \dots has least 2, as it repeats every two terms. Periodic sequences exhibit several key properties that make them analytically tractable. The sum of the terms over one full period determines the value of the sequence, and for sequences defined over integers, periodicity often arises in contexts. A related concept is the eventually periodic sequence, which becomes periodic after a finite initial segment but may not repeat from the start. In , periodic sequences play a crucial role in , such as in the Pisano periods of Fibonacci-like sequences a prime, which measure the length of repeating cycles in linear recurrences. They also appear in for studying binomial coefficients primes. Beyond pure mathematics, periodic sequences model repeating signals in engineering and signal processing, enabling applications like for waveform decomposition.

Fundamentals

Definition

In mathematics, a sequence \{a_n\}_{n=1}^\infty of complex numbers (or more generally, elements from any set) is said to be periodic with period p if a_{n+p} = a_n for all positive integers n \geq 1, where p is a positive integer. The smallest such positive integer p is called the minimal period or fundamental period of the sequence. From the periodicity condition, it follows that a_{n + k p} = a_n for all positive s n \geq 1 and all nonnegative s k \geq 0, obtained by repeated application of the defining relation. A periodic sequence in this sense is called purely periodic if the periodicity holds starting from the initial term n=1; in contrast, an eventually periodic sequence (or ultimately periodic sequence) satisfies the condition only for all n \geq m for some fixed m > 1, with a preperiod of length m-1. The full treatment of eventually periodic sequences, including their properties and examples, is addressed separately. Periodic sequences serve as discrete analogs of periodic functions, where a continuous function f: \mathbb{R} \to \mathbb{C} is periodic with period T > 0 if f(x + T) = f(x) for all x \in \mathbb{R}, mirroring the repetition in the sequence case but over the integers.

Examples

A constant sequence, defined by a_n = c for all positive integers n where c is a fixed value, exemplifies the simplest periodic sequence with minimal period 1, since a_{n+1} = a_n holds for every n. This repetition of the same term indefinitely satisfies the periodicity condition with the smallest possible shift. The alternating given by a_n = (-1)^n, which produces the terms -, , -, , ..., demonstrates periodicity with minimal 2, as a_{n+2} = a_n for all n, but no smaller positive works universally. Such sequences appear in contexts like discrete-time signals where sign alternation repeats every two steps. A basic repeating pattern , such as a_n = 1 if n is odd and a_n = 2 if n is even (yielding 1, 2, 1, 2, ...), has minimal 2, illustrating how non-constant values can cycle consistently. The decimal expansion of $1/7 = 0.\overline{142857} corresponds to the periodic sequence of digits 1, 4, 2, 8, 5, 7 repeating with minimal period 6, a classic example from rational number representations where the cycle length equals the multiplicative order of 10 modulo 7. Simple binary cycles, like the sequence 0, 1, 0, 1, ..., also exhibit minimal period 2 and serve as foundational examples in combinatorial mathematics. More generally, the sequence of powers \zeta^n, where \zeta is a primitive p-th root of unity (a complex number satisfying \zeta^p = 1 and no smaller positive exponent yields 1), is periodic with minimal period p, cycling through the p distinct roots before repeating.

Properties

Partial Sums

For a periodic sequence \{a_n\}_{n=1}^\infty with period p, the partial sums S_n = \sum_{i=1}^n a_i can be expressed explicitly by grouping terms into complete periods plus a remainder. Write n = kp + m where k = \lfloor n/p \rfloor and $0 \leq m < p. Due to periodicity, a_{jp + i} = a_i for j \geq 0 and $1 \leq i \leq p, so S_{kp + m} = \sum_{j=0}^{k-1} \sum_{i=1}^p a_{jp + i} + \sum_{i=1}^m a_{kp + i} = k \sum_{i=1}^p a_i + \sum_{i=1}^m a_i. Let \Sigma_p = \sum_{i=1}^p a_i. The second sum depends only on m and thus repeats every p steps in n. This derivation follows from partitioning . The behavior of \{S_n\} depends on \Sigma_p. If \Sigma_p = 0, then S_n = \sum_{i=1}^m a_i for n = kp + m, so the partial sums are bounded (taking at most p distinct values) and form a periodic sequence with period p. If \Sigma_p \neq 0, the partial sums grow linearly: S_n = (n/p) \Sigma_p + r_m, where r_m = \sum_{i=1}^m a_i - (m/p) \Sigma_p is a bounded periodic fluctuation with period p. An important implication is that the arithmetic mean of the sequence is \lim_{n \to \infty} S_n / n = \Sigma_p / p, which equals zero precisely when \Sigma_p = 0.

Products

The partial product of a periodic sequence \{a_n\} with period p is defined as P_n = \prod_{i=1}^n a_i. To analyze its behavior, group the terms into complete periods. For n = kp + m where $0 \leq m < p, the product consists of k full periods followed by the first m terms of the next period. Let \Pi_p = \prod_{i=1}^p a_i denote the product over one full period, and \Pi_m = \prod_{i=1}^m a_i. Then, P_{kp + m} = \Pi_p^k \cdot \Pi_m. This formula follows directly from the periodicity, as each full period contributes a factor of \Pi_p, and the remaining terms form the partial product \Pi_m, which is bounded since m < p. The long-term behavior of the partial products depends on the value of \Pi_p. If \Pi_p = 1, the sequence of partial products P_n is itself periodic with period p, because P_{kp + m} = 1^k \cdot \Pi_m = \Pi_m = P_m, repeating every p steps. If |\Pi_p| < 1, the partial products converge to 0 as n \to \infty (assuming no zeros in the sequence), since |\Pi_p|^k \to 0 and \Pi_m is bounded. Conversely, if |\Pi_p| > 1 and the sequence contains no zeros, the partial products grow exponentially, with |P_n| approximately |\Pi_p|^{n/p}, reflecting geometric growth at rate \log |\Pi_p| / p per term. However, if the periodic sequence includes a zero within the period (so \Pi_p = 0), then after the first full period containing the zero, all subsequent partial products are zero, leading to eventual constancy at zero. When the terms a_n > 0 for all n, the growth rate can be analyzed via logarithmic sums: \log P_n = \sum_{i=1}^n \log a_i. Since \{ \log a_n \} is also periodic with period p and sum over the period \log \Pi_p, the partial sum \log P_{kp + m} = k \log \Pi_p + \sum_{i=1}^m \log a_i, mirroring the additive case but multiplicatively. This yields an average growth rate of (\log \Pi_p)/p per term, useful for asymptotic analysis in positive-term periodic sequences.

Special Cases

Periodic 0-1 Sequences

Periodic 0-1 sequences, or periodic sequences, constitute a of periodic sequences in which each term a_n takes values in the set \{0, 1\}. These sequences repeat a fixed finite of length p (the period) indefinitely and are widely studied in for their properties, which facilitate applications in pseudorandom number generation, , and sequence design for communication systems. Representative examples include the constant sequence (1, 1, 1, \dots) with 1, where every term is 1, and the alternating sequence (1, 0, 1, 0, \dots) with 2, which exhibits maximal balance between 0s and 1s within each . More complex instances arise in constructions like periodic complementary sets, where multiple sequences of equal length have periodic autocorrelations summing to an ideal delta function. A fundamental property concerns partial sums: over one full period of length p, the sum \sum_{k=1}^p a_k equals the number of 1s in the repeating block, providing a direct measure of the sequence's "weight." The asymptotic density of 1s, defined as \lim_{N \to \infty} (1/N) \sum_{k=1}^N a_k, simplifies to this weight divided by p, yielding a rational value between 0 and 1 that characterizes the sequence's overall composition. For balanced sequences (density $1/2), the partial sums over multiples of the period deviate minimally from uniformity. The minimal period of a binary sequence is the smallest positive integer p such that a_{n+p} = a_n for all n \geq 1. Enumerating distinct such sequences up to cyclic shifts—equivalent to counting binary necklaces—relies on the necklace polynomial M_n(x) = \frac{1}{n} \sum_{d \mid n} \phi(d) x^{n/d}, where \phi is Euler's totient function; for binary sequences, evaluate at x=2 to obtain the count of necklaces of length n. This formula derives from the cycle index of the cyclic group acting on binary strings and factors prominently through cyclotomic polynomials \Phi_d(x), which capture the primitive contributions for divisors d of n. Primitive binary necklaces, corresponding to sequences with exact minimal period n, number \frac{1}{n} \sum_{d \mid n} \mu(d) 2^{n/d}, where \mu is the Möbius function, further highlighting the role of cyclotomic identities in decomposition. In discrepancy theory, periodic 0-1 sequences with balanced density serve as building blocks for low-discrepancy constructions, where the goal is to minimize the supremum deviation of partial sums from their expected uniform behavior. Such sequences ensure local balance, with the maximum imbalance over substrings (discrepancy) bounded relative to the period length, as seen in de Bruijn sequences that achieve near-optimal uniformity in 0-1 distributions.

Eventually Periodic Sequences

An eventually periodic sequence is a sequence that becomes periodic after a finite initial segment, called the preperiod. Formally, a sequence (a_n)_{n \geq 0} in a is eventually periodic with preperiod k \geq 0 and period p \geq 1 if a_{n+p} = a_n for all n \geq k. A prominent example arises in the decimal expansions of rational numbers, which are always eventually periodic. For instance, \frac{1}{6} = 0.1666\ldots, where the digit sequence after the point has a preperiod of 1 (the 1) followed by a repeating of 1 (the 6). The ordinary \sum_{n=0}^\infty a_n x^n of an eventually periodic is rational, meaning it can be expressed as the ratio of two polynomials. After the preperiod, the partial sums of the grow linearly at a rate equal to the value of the periodic part, with bounded fluctuations similar to those in strictly periodic s. Eventually periodic sequences are also closely tied to finite , as they can be generated and recognized by finite state machines that store the preperiod and through the period using finite memory. Unlike strictly periodic sequences, where a minimal period is the smallest p satisfying the equality for all n \geq 0, eventually periodic sequences with k > 0 lack a well-defined minimal period for the entire sequence, since the repetition does not hold throughout the preperiod.

Generalizations and Applications

Almost Periodic Sequences

Almost periodic sequences generalize periodic sequences by allowing approximate repetitions that occur with varying periods, without requiring exact matches at a fixed . Unlike strictly periodic sequences, which repeat identically after a fixed period, almost periodic sequences exhibit patterns that can be approximated arbitrarily closely by periodic ones over the entire line, with these approximations occurring densely. This concept was developed by in the 1920s as an extension of periodicity to capture more complex oscillatory behaviors in and . A bounded sequence \{a_n\}_{n \in \mathbb{Z}} with values in \mathbb{C} (or a Banach space) is Bohr almost periodic if, for every \varepsilon > 0, the set of \varepsilon-periods \{p \in \mathbb{Z} : \sup_n |a_{n+p} - a_n| < \varepsilon\} is relatively dense in \mathbb{Z}. Relatively dense means there exists l = l(\varepsilon) > 0 such that every interval of length l in \mathbb{Z} contains at least one such p. This ensures that the sequence "almost repeats" with good approximations to periodicity at irregular but dense intervals, distinguishing it from periodic sequences, which possess a single exact period p satisfying a_{n+p} = a_n for all n. The class of Bohr almost periodic sequences forms a vector space closed under uniform limits, addition, and scalar multiplication, and all such sequences are bounded. Bohr's Fourier characterization states that a sequence is almost periodic if and only if it is the uniform limit of trigonometric polynomials of the form \sum_{k=1}^m c_k \exp(2\pi i \lambda_k n), where c_k \in \mathbb{C} and \lambda_k \in \mathbb{R}. The frequencies \{\lambda_k\} form the of the sequence, and the coefficients c_k are uniquely determined as Bohr-Fourier coefficients via the mean value c_k = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N a_n \exp(-2\pi i \lambda_k n), which exists for almost periodic sequences. A related notion is Besicovitch almost periodicity, where sequences are limits in the Besicovitch semi-norm (mean-square sense) of such trigonometric polynomials, forming a broader class useful in L^2 settings like . For almost periodic sequences with zero mean (i.e., \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N a_n = 0), the partial sums \sum_{k=1}^N a_k = O(1) are bounded. Examples include the sequence a_n = \exp(2\pi i \alpha n) for irrational \alpha \in [0,1), which is almost periodic due to the dense orbit of \{n\alpha\} modulo 1 but not periodic since the frequencies do not commensurate. More complex instances arise in quasicrystals, where diffraction patterns correspond to almost periodic sequences modeling atomic arrangements, such as projections of higher-dimensional lattices onto lower dimensions, exhibiting long-range order without translational periodicity. These sequences appear in Sturmian words or Fibonacci substitutions, providing models for aperiodic tilings.

Applications

In , the expansions of are eventually periodic, meaning that after a finite number of digits, the sequence of decimal digits repeats indefinitely; this property distinguishes from irrationals, whose expansions are non-repeating. Similarly, in the system, every has an eventually periodic p-adic expansion, where the digits to the left of the p-adic point repeat after some initial terms, providing a characterization of in this topology. Periodic continued fractions arise in the expansions of quadratic irrationals, where the sequence of partial quotients repeats, enabling efficient approximations and connections to Pell equations for solving Diophantine problems. In dynamical systems, periodic sequences model periodic orbits, which are fixed points of iterated maps under composition; for instance, in the x_{n+1} = r x_n (1 - x_n), periodic orbits emerge for certain parameter values r, such as period-2 cycles for $3 < r < 1 + \sqrt{6}, illustrating bifurcations and the onset of chaos in discrete-time systems. algorithms, like Floyd's tortoise-and-hare method, identify periodic structures in sequences generated by functional iterations, such as detecting loops in linked lists or rho-shaped graphs with constant space and linear , by advancing two pointers at different speeds until they meet in the . In , discrete periodic sequences represent sampled periodic signals, which are analyzed using the (DFT) to decompose them into frequency components; the DFT assumes periodicity over N samples, enabling efficient computation via the (FFT) for applications like of repeating waveforms. The Nyquist-Shannon sampling theorem ensures that periodic bandlimited signals can be perfectly reconstructed from discrete samples if the sampling rate exceeds twice the highest frequency, preventing and linking periodic sequences to continuous-time representations in reconstruction. In , periodic sequences underpin string algorithms like the Knuth-Morris-Pratt (KMP) , where the prefix function computes the longest proper prefix that is also a for each , exploiting periodicity (or borders) to skip redundant comparisons and achieve linear-time search for patterns in texts. Cyclic codes, a class of linear invariant under cyclic shifts, are widely used in error-correcting applications such as Reed-Solomon codes for data transmission; their generator polynomials correspond to periodic sequences in the , allowing efficient encoding and decoding of errors in cyclic redundancy checks. Periodic 0-1 sequences with low discrepancy, such as those derived from van der Corput or Halton constructions, serve as quasirandom generators in simulations, mimicking while minimizing clustering to improve convergence rates over truly random binary sequences. Šindel sequences, which are periodic sequences designed such that their partial sums exactly match the triangular numbers T_k = k(k+1)/2, connect periodicity to combinatorial identities and have applications in modeling the mechanics of Prague's , where the sequence periods align with gear ratios for triangular progressions.