When does \(F^L_m\) divide \(F_n\)? A combinatorial solution (Q2883384)

Let \(\{F_i \}_{i \geq 1}\) be a Fibonacci sequence. Let for \(m, r \geq 1\), \(r_0 = 0\) and for \(k \geq 1\), NEWLINE\[NEWLINEr_k = F_k \binom{r}{k} + \frac{F_{m-1}}{F_m} r_{k-1}.NEWLINE\]NEWLINE Some theorems are proved, namely NEWLINENEWLINETheorem 3. For \(m, r \geq 0\), NEWLINE\[NEWLINEF_{mr} = \sum_{j=1}^r \binom{r}{k} F_j F^j_m F^{r-j}_{m-1}.NEWLINE\]NEWLINE NEWLINENEWLINETheorem 5. For \(L, m, r \geq 1\): \(F^L_m\mid F_{mr}\) if and only if \(F_m\mid r_{L-1}\).NEWLINENEWLINE NEWLINELet \(u_0 = 0, u_1 = 1\), and for \(n \geq 2\), \(u_n = au_{n-1} + bu_{n-2}\). NEWLINENEWLINETheorem 6. For \(m, r \geq 0\), NEWLINE\[NEWLINEu_{mr} = \sum_{j=1}^r \binom{r}{k} u_j u^j_m b^{r-j}u^{r-j}_{m-1}.NEWLINE\]