For a positive integer \(n\), an \(n\)-sequence is a sequence \((a_0,\ldots,a_n)\) of non-negative integers satisfying the following condition: if \(i\) and \(j\) are non-negative integers with \(i+j \leqslant n\), then \(a_i+a_j \leqslant n\) and \(a_{a_i+a_j}=a_{i+j}\).

Let \(f(n)\) be the number of \(n\)-sequences. Prove that there exist positive real numbers \(c_1\), \(c_2\), and \(\lambda\) such that \(c_1\lambda^n < f(n) < c_2\lambda^n\) for all positive integers \(n\).