Tetrahedral number

A tetrahedral number, also called a triangular pyramidal number, is a figurate number that counts the points needed to build a pyramid with a triangular base and three triangular sides, a tetrahedron. The nth tetrahedral number is the sum of the first n triangular numbers, and it has the closed form n(n+1)(n+2)/6, which is the same as the binomial coefficient of n+2 choose 3. The sequence begins 1, 4, 10, 20, 35, 56, 84, 120, 165, 220 and appears in the fourth diagonal of Pascal's triangle. Because the formula factors into three consecutive integers over six, no tetrahedral number above 1 can be prime. Meyl proved in 1878 that only three of them are perfect squares, 1, 4 and 19600, and the numbers are linked to Pollock's conjecture that every positive integer is a sum of at most five of them. They can be pictured as spheres stacked in a tetrahedral pile, as with a rack of billiard balls topped by successive layers.