Perturbing the Conway sequence

We study the alternating perturbation of Conway’s nested recurrence, $C(n)=C(C(n-1))+C(n-C(n-1))+(-1{)}^n$ for $n\ge 3$, with $C(1)=C(2)=1$. We prove that the sequence is well defined and satisfies $C(n)={\psi }^{-3}n+O(n\mathrm{/}\sqrt{\log n})$, where $\psi >1$ is the real root of ${\psi }^3={\psi }^2+1$.