Centered tetrahedral number

Centered tetrahedral number
Total no. of terms	Infinity
Subsequence of	Polyhedral numbers
Formula
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.^[1] The $n$ th centered tetrahedral number, starting at $n=0$ for a single dot, is:^[2]^[3]

\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

^ ^a ^b Deza, E.; Deza, M. (2012). Figurate Numbers. Singapore: World Scientific Publishing. pp. 126–128. ISBN 978-981-4355-48-3.
^ ^a ^b Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Deza numbers the centered tetrahedral numbers at $n=1$ for a single dot, leading to a different formula.

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[deza-1] Deza, E.; Deza, M. (2012). Figurate Numbers. Singapore: World Scientific Publishing. pp. 126–128. ISBN 978-981-4355-48-3.

[oeis-2] Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[3] Deza numbers the centered tetrahedral numbers at $n=1$ for a single dot, leading to a different formula.

[1]

[2]

[3]