Category: Recurrence relations

Integrodifference equation
In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: where is a sequence in the function space and is the domain of those functions. In mo
Somos sequence
In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their definin
Lucas sequence
In mathematics, the Lucas sequences and are certain constant-recursive integer sequences that satisfy the recurrence relation where and are fixed integers. Any sequence satisfying this recurrence rela
Linear recurrence with constant coefficients
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equatio
Skolem–Mahler–Lech theorem
In additive and algebraic number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers satisfies a linear difference equation, then with finitely many exceptions the positions at
K-regular sequence
In mathematics and theoretical computer science, a k-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-k representations of the integers. The class of k-regul
McCarthy 91 function
The McCarthy 91 function is a recursive function, defined by the computer scientist John McCarthy as a test case for formal verification within computer science. The McCarthy 91 function is defined as
Jacobsthal number
In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence for
Initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time
Rational difference equation
A rational difference equation is a nonlinear difference equation of the form where the initial conditions are such that the denominator never vanishes for any n.
Juggler sequence
In number theory, a juggler sequence is an integer sequence that starts with a positive integer a0, with each subsequent term in the sequence defined by the recurrence relation:
Pell number
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This se
Keith number
In number theory, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number in a given number base with digits such that when a sequence is created such that th
Akra–Bazzi method
In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorith
Skolem problem
In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over differe
Sylvester's sequence
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are 2, 3, 7, 43, 18
Primefree sequence
In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fi
Lucas number
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci num
Recurrence relation
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the seque
Difference Equations: From Rabbits to Chaos
Difference Equations: From Rabbits to Chaos is an undergraduate-level textbook on difference equations, a type of recurrence relation in which the values of a sequence are determined by equations invo
Recamán's sequence
In mathematics and computer science, Recamán's sequence is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, t
Perrin number
In mathematics, the Perrin numbers are defined by the recurrence relation P(n) = P(n − 2) + P(n − 3) for n > 2, with initial values P(0) = 3, P(1) = 0, P(2) = 2. The sequence of Perrin numbers starts
Jacobi operator
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonorm
Constant-recursive sequence
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or mor
Ménage problem
In combinatorial mathematics, the ménage problem or problème des ménages asks for the number of different ways in which it is possible to seat a set of male-female couples at a round dining table so t
Time-scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite
Padovan sequence
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values and the recurrence relation The first few values of P(n) are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16,
Göbel's sequence
In mathematics, Göbel's sequence is a sequence of rational numbers defined by the recurrence relation with starting value Göbel's sequence starts with 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... (s
Matrix difference equation
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous poin
Master theorem (analysis of algorithms)
In the analysis of algorithms, the master theorem for divide-and-conquer recurrences provides an asymptotic analysis (using Big O notation) for recurrence relations of types that occur in the analysis
Leonardo number
The Leonardo numbers are a sequence of numbers given by the recurrence: Edsger W. Dijkstra used them as an integral part of his smoothsort algorithm, and also analyzed them in some detail. A Leonardo