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A sequence is constant-recursive of order less than or equal to if and only if it can be written as. where is a vector, is a matrix, and is a vector, where the elements come from the same domain (integers, rational numbers, algebraic numbers, real numbers, or complex numbers) as the original sequence.
A recursive step — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ancestor. One's ancestor is either: One's parent (base case), or; One's parent's ancestor (recursive step). The Fibonacci sequence is another classic example of recursion: Fib(0) = 0 as ...
Recursion (computer science) Tree created using the Logo programming language and relying heavily on recursion. Each branch can be seen as a smaller version of a tree. Recursive drawing of a Sierpiński Triangle through turtle graphics. In computer science, recursion is a method of solving a computational problem where the solution depends on ...
A linear recurrence with constant coefficients is an equation of the following form, written in terms of parameters a1, ..., an and b: or equivalently as. The positive integer is called the order of the recurrence and denotes the longest time lag between iterates. The equation is called homogeneous if b = 0 and nonhomogeneous if b ≠ 0.
A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form. where. is a function, where X is a set to which the elements of a sequence must belong.
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 relation can be represented as a linear combination of the Lucas sequences and. More generally, Lucas sequences and represent ...
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from ...
In mathematics and computer science, Recamán's sequence[1][2] is a well known sequence defined by a recurrence relation. Because its elements are related to the previous elements in a straightforward way, they are often defined using recursion. It takes its name after its inventor Bernardo Recamán Santos [es], a Colombian mathematician.