Triangular Number - Relations To Other Figurate Numbers

Relations To Other Figurate Numbers

Triangular numbers have a wide variety of relations to other figurate numbers.

Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically,

Alternatively, the same fact can be demonstrated graphically:

There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36. Some of them can be generated by a simple recursive formula:

with

All square triangular numbers are found from the recursion

with and

Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n.

The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number,

More generally, the difference between the nth m-gonal number and the nth (m + 1)-gonal number is the (n - 1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number: the nth centered k-gonal number is obtained by the formula

where T is a triangular number.

The positive difference of two triangular numbers is a trapezoidal number.