Recognition criterion for triangular numbers ($M=T_n⟺8M+1=(2n+1{)}^2$)

A nonnegative integer $M$ is a triangular number exactly when $8M+1$ is a perfect square. Its square root is then an odd integer, and the row index isCompleting the square in $M=n(n+1)/2$ proves necessity; reversing the calculation proves sufficiency. For example, $8\cdot 21+1=169=13^2$, giving $n=6$. In contrast, $8\cdot 20+1=161$ lies strictly between $12^2$ and $13^2$, so twenty is not a triangular number. The criterion turns an apparently open-ended search through a sequence into one exact perfect square check.