= 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 $u_n=u_n^{\mathrm p}+Av_n+Bw_n$ are determined by $u_j,u_k$ exactly when $v_jw_k-v_kw_j\ne0$. 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> $r\ne0$, use $v_n=r^n$, $w_n=nr^n$. The determinant is $(k-j)r^{j+k}\ne0$ for distinct $j,k$, so any such pair of prescribed values determines the two coefficients.
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