Closed Form for Sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...

Theorem
Let $a_1, a_2, a_3, \ldots$ be the sequence:
 * $\left \langle{a_n}\right \rangle = 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots$

Then:
 * $a_n = \left \lceil{\dfrac {\sqrt {1 + 8 n} - 1} 2}\right \rceil$

Proof
From Closed Form for Triangular Numbers, for $n = 1, 3, 6, 10, \ldots$:
 * $n = \dfrac {a_n \left({a_n + 1}\right)} 2$

Thus by the Quadratic Formula: $a_n = \dfrac {-1 \pm \sqrt {1 + 8 n} } 2$

In this context it is the positive root that is required.

The result follows by definition of ceiling function.