Polygonal numbers represent an ancient link between geometry and number theory, and can be traced back to the Greeks (Pythagoras and his followers, in fact).
A good question is, “Can we find a formula for the nth polygonal number?”
A good answer is, “Yes.” To do so requires the so-called “method of finite differences”. Well, it doesn’t really “require” it…
Some Properties of Polygonal Numbers (that the Greeks knew about)
Twice any triangular number is an oblong number.
The nth square number exceeds its predecessor by the sum of their two roots (Fibonacci).
The nth square number is the sum of the first n odd integers (Pythagoras).
Eight times any triangular number plus one is a square number (Pythagoras).
Any square number is the sum of two consecutive triangular numbers (Nichomachus, Introduction to Arithmetic).
Any pentagonal number is the sum of a square number and triangular number (Nichomachus).
There exists an “odd number triangle cubic sum mystery”.
Fermat once conjectured that every positive integer is the sum of at most 3 triangular numbers, or at most 4 square numbers, etc…
Final Thoughts on Polygonal Numbers
Find a formula for the nth m-gonal number .
Show that if denotes the nth triangular number, then the following numbers are also triangular: . Can you generalize?