Sum of Floor Square Roots

$a_n = \lfloor\sqrt{n}\rfloor$. Group $n$ based on values of $\lfloor\sqrt{n}\rfloor = k$ (which occurs for $2k+1$ values of $n$):

• $k = 1$: $n \in [1, 3]$ (3 values) $\implies 3 \times 1 = 3$
• $k = 2$: $n \in [4, 8]$ (5 values) $\implies 5 \times 2 = 10$
• $k = 3$: $n \in [9, 15]$ (7 values) $\implies 7 \times 3 = 21$
• $k = 4$: $n \in [16, 24]$ (9 values) $\implies 9 \times 4 = 36$
• $k = 5$: $n \in [25, 35]$ (11 values) $\implies 11 \times 5 = 55$
• $k = 6$: $n \in [36, 48]$ (13 values) $\implies 13 \times 6 = 78$
• $k = 7$: $n = 49, 50$ (2 values) $\implies 2 \times 7 = 14$

$\text{Total sum} = 3 + 10 + 21 + 36 + 55 + 78 + 14 = 217$