Centered tetrahedral number

Centered tetrahedral number

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

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]}

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

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

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

References

1. Deza, E.; Deza, M. (2012). Figurate Numbers. Singapore: World Scientific Publishing. pp.  126–128. ISBN   978-981-4355-48-3.
2. Sloane, N. J. A. (ed.). "SequenceA005894(Centered tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. Deza numbers the centered tetrahedral numbers at ${\displaystyle n=1}$ for a single dot, leading to a different formula.