mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a centered tetrahedral number is a centered figurate number that represents a

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

. That is, it counts the dots in a three-dimensional dot pattern with a single dot surrounded by tetrahedral shells. The

n

th centered tetrahedral number, starting at

n=0

for a single dot, is:Deza numbers the centered tetrahedral numbers at

n=1

for a single dot, leading to a different formula. The first such numbers are:

References

Figurate numbers {{Num-stub