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triangular number

Any number that can be represented by a triangular array of dots. 1, 3, 6, 10, ... are triangular numbers. The nth triangular number is n(n + 1)/2. Every integer is the sum of at most three triangular numbers. Every even triangular number is a perfect number. If T is a triangular number, 8T + 1 is a perfect square and 9T + 1 is another triangular number. The square of the nth triangular number is equal to the sum of the first n cubes. Certain triangular numbers are also squares, but no triangular number can be a third, fourth or fifth power, nor can one end in 2, 4, 7, or 9.

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