Putnam 1983 B2

For positive integers $n$ , let $C(n)$ be the number of representation of $n$ as a sum of nonincreasing powers of $2$ , where no power can be used more than three times. For example, $C(8)=5$ since the representations of $8$ are: $$8,4+4,4+2+2,4+2+1+1,\text{ and }2+2+2+1+1.$$ Prove or disprove that there is a polynomial $P(x)$ such that $C(n)=\lfloor P(n)\rfloor$ for all positive integers $n$ .