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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.
In mathematics, an infinite sequence of numbers is called constant-recursive if it satisfies an equation of the form. for all , where are constants. The equation is called a linear recurrence relation . The concept is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence.
Generalizations. The Catalan numbers can be interpreted as a special case of the Bertrand's ballot theorem. Specifically, is the number of ways for a candidate A with n + 1 votes to lead candidate B with n votes. The two-parameter sequence of non-negative integers is a generalization of the Catalan numbers.
Sum ← Sum + x. SumSq ← SumSq + x × x. Var = (SumSq − (Sum × Sum) / n) / (n − 1) This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to ...
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 simple formula is. for positive integer , where is the floor function, which rounds down to the nearest integer. By Wilson's theorem, is prime if and only if . Thus, when is prime, the first factor in the product becomes one, and the formula produces the prime number . But when is not prime, the first factor becomes zero and the formula ...
Generating function. In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Unlike an ordinary series, the formal power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an ...