Centered tetrahedral number

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.^[1] The ${\displaystyle n}$th centered tetrahedral number, starting at ${\displaystyle n=0}$ for a single dot, is:^[2][3]

Key Information

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

The first such numbers are:^[1][2]

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