A class of sequences defined by nonlinear recurrences involving the greatest integer function \([.]\) is studied, a typical member of the class being \(a(0)=1\), \(a(n)=a([n/2])+a([n/3])+a([n/6])\) for \(n\geq 1\). For this sequence, it is shown that \(\lim a(n)/n\) as \(n\to \infty\) exists and equals \(12/(log 432)\). More generally, for any sequence defined by \(a(0)=1\), \(a(n) = \sum^{s}_{i=1} r_ia([n/m_i])\) for \(n\geq 1\), where \(r_i>0\) and the \(m_i\) are integers \(\geq 2\), the asymptotic behavior of \(a(n)\) is determined. Let \(\tau\) be the unique solution to \(\sum^{s}_{i=1} r_im_i^{-\tau} = 1\). When there is an integer \(d\) and integers \(u_i\) such that \(m_i=d^{u_i}\) for all \(i\), \(a(n)/n^{\tau}\) oscillates, while in the other case, where no such d and \(u_i\) exist, the limit of \(a(n)/n^{\tau}\) exists and is explicitly computed. Results on the speed of convergence to the limit are also obtained.