Octahedral number

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

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

The first few octahedral numbers are:

1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891.

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

  • 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]
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