constant-recursive sequence

Article summary

of some subclasses of constant-recursive sequences, ordered by inclusion In mathematics, an infinite sequence of numbers s_0, s_1, s_2, s_3, \ldots is called constant-recursive if it satisfies an equation of the form : s_n = c_1 s_{n-1} + c_2 s_{n-2} + \dots + c_d s_{n-d}, for all n \ge d , where c_i 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. For example, the Fibonacci sequence : 0, 1, 1, 2, 3, 5, 8, 13, \ldots , is constant-recursive because it satisfies the linear recurrence F_n = F_{n-1} + F_{n-2} : each number in the sequence is the sum of the previous two. Other examples include the power of two sequence 1, 2, 4, 8, 16, \ldots , where each number is the sum of twice the previous number, and the square number sequence 0, 1, 4, 9, 16, 25, \ldots . All arithmetic progressions, all geometric progressions, and all polynomials are constant-recursive. However, not all sequences are constant-recursive; for example, the factorial sequence 1, 1, 2, 6, 24, 120, \ldots is not constant-recursive. Constant-recursive sequences are studied in combinatorics and the theory of finite differences.…

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