Pentagonal numbers are truncated triangular numbers
johndcook.comJohn2026年04月01日 13:23
Pentagonal numbers are truncated triangular numbers. You can take the diagram that illustrates the nth pentagonal number and warp it into the base of the image that illustrates the (2n − 1)st triangular number. If you added a diagram for the (n − 1)st triangular number to the bottom of the image on the right, you’d have a diagram for the (2n − 1)st triangular number.
In short,
Pn = T2n − 1 − Tn.
This is trivial to prove algebraically, though the visual proof above is more interesting.
The proof follows immediately from the definition of pentagonal numbers
Pn = (3n² − n)/2
and triangular numbers
Tn = (n² − n)/2.