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Octahedral Number

An octahedral number is a figurate number which is the sum of two consecutive square pyramidal numbers,

O_n=P_(n-1)+P_n=1/3n(2n^2+1).
(1)

Here n is a positive integer, P_n is the nth square pyramidal number, and P_0=0. The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, ... (OEIS A005900). The generating function for the octahedral numbers is

(x(x+1)^2)/((x-1)^4)=x+6x^2+19x^3+44x^4+...
(2)

for |x|<1.

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

HauyOctahedron03
HauyOctahedron05
HauyOctahedron07
HauyOctahedron09
HauyOctahedron11

A related set of numbers is the number of cubes in the Haüy construction of the octahedron. Each cross section has area

S_n=n+2sum_(i=1,3,...,n-2)i=1/2(n^2+1),
(3)

where n is a positive odd number, and adding all cross sections gives

HO_k=S_k+2sum_(i=1,3,...,k-2)S_i=1/6k(k^2+5)
(4)

for k a positive odd number. Re-indexing so that k=2n-1 gives

HO_n=1/3(2n-1)(2n^2-2n+3),
(5)

the first few values of which are 1, 7, 25, 63, 129, ... (OEIS A001845). These numbers have the generating function

f(x)=sum_(n=1)^inftyHO_nx^(n-1)=((1+x)^3)/((1-x)^4)=1+7x+25x^2+63x^3+129x^4+...
(6)

for |x|<1.

See also

Haüy Construction, Octahedron, Truncated Octahedral Number

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References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 50, 1996.Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005.Pollock, F. "On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders." Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.Sloane, N. J. A. Sequences A001845/M4384 and A005900/M4128 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Octahedral Number

Cite this as:

Weisstein, Eric W. "Octahedral Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OctahedralNumber.html

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