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

A centered tetrahedral number is a centered figurate number that represents a tetrahedron. The centered tetrahedral number for a specific n is given by

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

The first such numbers are 1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, ... (sequence A005894 in the OEIS).

Parity and divisibility

• Every centered tetrahedral number is odd.
• Every centered tetrahedral number with an index of 2, 3 or 4 modulo 5 is divisible by 5.
• The only prime centered tetrahedral number is 5. We only need to check when either ${\displaystyle 2n+1}$ or ${\displaystyle n^{2}+n+3}$ is a divisor of 3.

References

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