# Difference between revisions of "Recursive sequence"

recurrent sequence

A sequence \$a_0,a_1,\ldots,\$ that satisfies a relation

\$\$a_{n+p}+c_1a_{n+p-1}+\ldots+c_pa_n=0,\$\$

where \$c_1,\ldots,c_p\$ are constants. The relation permits one to compute the terms of the sequence one by one, in succession, if the first \$p\$ terms are known. A classical example of such a sequence is the Fibonacci sequence \$1,1,2,3,5,8\$ (\$a_{n+2}=a_{n+1}+a_n\$, \$a_0=a_1=1\$). A recursive series is a power series \$a_0+a_1x+a_2x^2+\ldots\$ whose coefficients form a recursive sequence. Such a series represents an everywhere-defined rational function.