centered tetrahedral number

{{Short description|Centered figurate number representing a tetrahedron}}

{{Infobox integer sequence

| number = Infinity

| parentsequence = Polyhedral numbers

| formula = $\backslash frac\{(2n+1)\backslash ,(n^2+n+3)\}\{3\}$

| first_terms = 1, 5, 15, 35, 69, 121, 195

| OEIS = A005894

| OEIS_name = Centered tetrahedral

}}

In mathematics, a centered tetrahedral number is a centered figurate number that represents a tetrahedron. That is, it counts the dots in a three-dimensional dot pattern with a single dot surrounded by tetrahedral shells.{{r|deza}} The $n$th centered tetrahedral number, starting at $n=0$ for a single dot, is:{{r|oeis}}Deza numbers the centered tetrahedral numbers at $n=1$ for a single dot, leading to a different formula.

{{bi|left=1.6|$\backslash displaystyle\; (2n+1)\backslash times\{(n^2+n+3)\; \backslash over\; 3\}.$}}

The first such numbers are:{{r|deza|oeis}}

{{bi|left=1.6|1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, ...}}