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 $n\geq 3$, with $C(1)=C(2)=1$. We prove that the sequence is well defined and satisfies $C(n)=ψ^{-3}n+O(n/\sqrt{\log n})$, where $ψ>1$ is the real root of $ψ^3=ψ^2+1$.