If n is Triangular then so is (2m+1)^2 n + m(m+1)/2

Theorem
Let $n$ be a triangular number.

Let $m \in \Z_{\ge 0}$ be a positive integer.

Then $\paren {2 m + 1}^2 n + \dfrac {m \paren {m + 1} } 2$ is also a triangular number.

Proof
Let $n$ be a triangular number.

Then:
 * $\exists k \in \Z: n = \dfrac {k \paren {k + 1} } 2$

So:

which is of the form:
 * $\dfrac {t \paren {t + 1} } 2$

where $t = \paren {2 m + 1} k + m$.