Square triangular number
= Square triangular number
{title2=$T_n=m^2$}
{wiki}
A <square triangular number> is simultaneously a <triangular number> and a <perfect square>. For example, $36=T_8=6^2$. This differs from a <squared triangular number>: $T_4^2=100$ is not itself triangular, by the <recognition criterion for triangular numbers>. Writing $T_n=m^2$ gives $(2n+1)^2-8m^2=1$, a <Pell equation> that explains why this intersection of two counting patterns has additional arithmetic structure.