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:
| 6 + 10 = 16 | 10 + 15 = 25 |
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.
Read more about this topic: Triangular Number
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