Well explained by Tom, from Cottenham
Village College

The answer I got for $T_n$ is: $8T_n+1=(2n+1)^2$

The $2n+1$ part is because the diagram looks like this for
$T_3$

$2T_n$ form a rectangle $n$ by $n+1$

The four rectangles rotate around the centre and together make
a square of side $n+(n+1)$

So we get the equation $8T_n+1=(2n+1)^2$