Every non-negative integer can be expressed that way: We write $ 4n+1$ as sum of three squares (which is possible by the Three-Squares-Theorem as $ 4n+1\not\equiv 0,4,7\mod 8$): $ 4n+1 = x^{2}+y^{2}+z^{2}$. Looking $ \mod 4$ (the squares $ \mod 4$ are only $ 0$ and $ 1$), we see that exactly two of $ x,y,z$ are even, w.l.o.g. $ x=2a$, $ y=2b$ and $ z=2c+1$ for some integers $ a,b,c$. We now have $ 4n+1 = (2a)^{2}+(2b)^{2}+(2c+1)^{2}= 4a^{2}+4b^{2}+4c^{2}+4c+1$, equivalent to $ n=a^{2}+b^{2}+c^{2}+c$.