*n*-th tetrahedral number (see OEIS A000292) represents the sum of the first

*n*-triangular numbers. In other words, the

*n*-th tetrahedral number represents the number of spheres in a tetrahedron (i.e. a triangular pyramid) each edge of which consists of

*n*spheres.

Let

*a(n)*be the

*n*-th tetrahedral number. One can easily establish that

*8*a(n) + (n+1) = a(2*n+1)*

*27*a(n) + 4*(n+1) = a(3*n+2)*

64*a(n) + 10*(n+1) = a(4*n+3) etc.

64*a(n) + 10*(n+1) = a(4*n+3) etc.

This is how one finds that,

*a(n)*(m+1)^3 + a(m)*(n+1) = a(m*n+m+n)*,

for any nonnegative integers

*m*and

*n.*