During my investigation on order of monomials (came from an $I$-adic topology on $\mathbb{F}_2[x]$) I encounter the recurrence relation $$c_{2n+1}=n^2+c_{n+1},\\ c_{2n}=n^2-n+1+c_n.$$ Unfortunately, I have no idea, how to proceed, well I never face some thing like that before.
Any hints or clues is appreciated.
Regards.