Solution of a linear recurrence

Definition

Let $R$ be a commutative ring, and let $E$ be a linear recurrence with constant coefficients over $R$, of order $d$ with coefficients $c_0, \dots, c_{d-1}$. A solution of $E$ is a sequence $u : \mathbb{N} \to R$ such that $$u_{n+d} = \sum_{i=0}^{d-1} c_i\, u_{n+i} \quad \text{for all } n \in \mathbb{N}.$$

Remarks

Each choice of the initial values $u_0, \dots, u_{d-1}$ gives exactly one solution. The sum of two solutions is a solution, and a multiple of a solution is a solution. So the solutions form the space of solutions. For $u_{n+2} = u_{n+1} + u_n$, the Lucas numbers $2, 1, 3, 4, 7, 11, \dots$ form a solution.

Explanations

L1 human

A solution is a sequence that obeys the rule of the recurrence at every step. The rule alone does not fix the sequence: you also choose the first $d$ terms.