$$ S = \sum_{k=1}^{n} \lfloor \sqrt{k}\rfloor. $$

Hello, I´m trying to solve this summation. I was able to get $$ a_n = 2a_{n-1} - a_{n-2} $$ for non perfect square numbers and $$ a_n = 2a_{n-1} - a_{n-2} + 1 $$ for perfect square numbers. But I´m clueless how to continue. I was thinking about some function which is 1 for square numbers and 0 for others and and make only one recurrence. But I don´t think that´s the way...


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