Series Solutions · Hard problem

For $y'' + ky' + y = 0$ with $k>0$, the power-series solution's coefficients come from a recurrence built on the characteristic roots of $r^2+kr+1=0$. The solution oscillates (coefficients behave like a decaying sine/cosine) exactly when those roots are complex. What is the smallest positive integer $k$ for which the solution is NOT oscillatory?