MathDB

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

B2