Let t 0 , t 1 , t 2 , ⋯ {t_0},{t_1},{t_2}, \cdots be a sequence of elements of a field F. We give a continued fraction algorithm for t 0 x + t 1 x 2 + t 2 x 3 + ⋯ {t_0}x + {t_1}{x^2} + {t_2}{x^3} + \cdots . If our sequence satisfies a linear recurrence, then the continued fraction algorithm is finite and produces this recurrence. More generally the algorithm produces a nontrivial solution of the system \[ ∑ j = 0 s t i + j λ j , 0 ⩽ i ⩽ s − 1 , \sum \limits _{j = 0}^s {{t_{i + j}}{\lambda _j},\quad 0 \leqslant i \leqslant s - 1,} \] for every positive integer s.