Centered tetrahedral number

Centered tetrahedral number

Total no. of terms	Infinity
Subsequence of	Polyhedral numbers
Formula	${\frac {(2n+1)\,(n^{2}+n+3)}{3}}$
First terms	1, 5, 15, 35, 69, 121, 195
OEIS index	A005894 Centered tetrahedral

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. The $n$th centered tetrahedral number, starting at $n=0$ for a single dot, is:

$\displaystyle (2n+1)\times {(n^{2}+n+3) \over 3}.$

The first such numbers are:

1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, ...