Octahedral number

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An octahedral number is a figurate number that represents an octahedron, or two square pyramids placed together, one upside-down underneath the other. The nth octahedral number $O n$ can be obtained by adding the n-1th and nth square pyramidal numbers together, or by using the following formula:

$O_n={1 \over 3}(2n^3 + n).$

The first few octahedral numbers are:

1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 (sequence A005900 in OEIS).

The octahedral numbers have a generating function

$\frac{z(z+1)^2}{(z-1)^4} = \sum_{n=1}^{\infty} O_n z^n = z +6z^2 + 19z^3 + \cdots .$

Sir Frederick Pollock conjectured in 1850 that every number is the sum of at most 7 octahedral numbers (Dickson 2005, p. 23): see Pollock octahedral numbers conjecture.

If $O n$ is the nth octahedral number and $T n$ is the nth tetrahedral number then

O n + 4 T n - 1 = T 2 n - 1 .

[edit] References

Dickson, L. E., History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005.
Eric W. Weisstein. "Octahedral Number." From MathWorld--A Wolfram Web Resource.[1]