Nonconsecutive initial values for a linear recurrence

A second-order linear recurrence relation normally uses two consecutive initial conditions, but two specified nonconsecutive values may also determine a unique solution of the linear recurrence relation. For two independent homogeneous solutions of the linear recurrence relation $v_n,w_n$, the coefficients in $u_n=u_n^{\mathrm{p}}+A{v}_n+B{w}_n$ are determined by $u_j,u_k$ exactly when $v_j{w}_k-v_k{w}_j\ne 0$. Otherwise the data can be inconsistent or can leave freedom. A nonconsecutive subscript is therefore not by itself evidence of an error.

For a repeated characteristic root of a linear recurrence $r\ne 0$, use $v_n=r^n$, $w_n=n{r}^n$. The determinant is $(k-j)r^{j+k}\ne 0$ for distinct $j,k$, so any such pair of prescribed values determines the two coefficients.