On \(r\)-Regular Compositions

If the integer \(r \geq 2\), say that a composition of the natural number \(n\) is \(r\)-\emph{regular} if no part is divisible by \(r\). Let \(c_r(n)\) denote the number of \(r\)-regular compositions of \(n\) (with \(c_r(0) = 1\)). We show that \(c_r(n)\) satisfies a linear recurrence of order \(r\). We also obtain asymptotic estimates for \(c_r(n)\), and we evaluate \(c_r(n)\) for \(2 \leq r \leq 5\) and \(1 \leq n \leq 10\).