### A000217 - Triangular numbers

A000217 - Triangular numbers

0, 1, 3, 6, 10, 15, 21, …

Triangular numbers appear in at least two very common games. In bowling, the pins are arranged into rows of 1, 2, 3, and 4 pins each, depicting T_{4} = 10. When setting up eight-ball on a billiard table, the triangular rack arranges the balls into T_{5} = 15. The largest repdigit triangular number is T_{36} = 666. Not only is 36 = 6^{2}, but it’s also triangular itself (T_{8} = 36). All even perfect numbers are triangular. Amazingly, they all have a prime index (i.e. T_{7} = 28).

Fermat’s Last Theorem is only the most well-known case where he claimed to have a solution to a problem but never published it. In 1638, he claimed to have a proof showing that every positive integer was equal to the sum of three triangular numbers, or four square numbers, or five pentagonal numbers, etc. A proof for the triangular case wasn’t published until Gauss did so 130 years after Fermat’s death. Jacobi and Lagrange each independently devised proofs of the square case around the same time. But it wasn’t until 1813 — almost 150 years after Fermat died — that his Polygonal Number Theorem was finally proved in its entirety.