Pentagonal numbers are truncated triangular numbers
by John from John D. Cook on (#74N74)
Pentagonal numbers are truncated triangular numbers. You can take the diagram that illustrates thenth 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 = (3n^2 - n)/2
and triangular numbers
Tn = (n^2 - n)/2.
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