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tetrahedral number

A number that can be made by considering a tetrahedral pattern of beads in three dimensions. For example, if a triangle of beads is made with three beads to a side, and on top of this is placed a triangle with two beads to a side, and on top of that a triangle with one bead to a side, the result is a tetrahedron of beads. In this case the total number of beads is (3rd triangular number) + (2nd triangular number) + (1st triangular number) = 6 + 3 + 1 = 10. In general the nth tetrahedral number is equal to the sum of the first n triangular numbers. This is the same as the 4th number from the left in the (n + 3)th row of Pascal's triangle. We can use the binomial formula for numbers in Pascal's triangle to show that the nth tetrahedral number is n+2C3, or (n + 2)(n + 1)n/6. The only numbers that are both tetrahedral and square are 4 (= 22 = T2) and 19600 (= 1402 = T48).

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