Editing Octahedral number (section)

==Properties and applications==
The octahedral numbers have a [[generating function]]

:<math> \frac{z(z+1)^2}{(z-1)^4} = \sum_{n=1}^{\infty} O_n z^n = z +6z^2 + 19z^3 + \cdots .</math>

[[Sir Frederick Pollock, 1st Baronet|Sir Frederick Pollock]] conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers.<ref>{{citation|author-link=L. E. Dickson|last=Dickson|first=L. E.|series=[[History of the Theory of Numbers]]|volume=2|title=Diophantine Analysis|location=New York|publisher=Dover|year=2005|pages=22–23|isbn=9780821819357 |url=https://books.google.com/books?id=eNjKEBLt_tQC&pg=PA22}}.</ref> This statement, the [[Pollock octahedral numbers conjecture]], has been proven true for all but finitely many numbers.<ref>{{citation|last=Elessar Brady|first=Zarathustra|arxiv=1509.04316|doi=10.1112/jlms/jdv061|issue=1|journal=Journal of the London Mathematical Society|mr=3455791|pages=244–272|series=Second Series|title=Sums of seven octahedral numbers|volume=93|year=2016|s2cid=206364502 }}</ref>

In [[chemistry]], octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called [[Magic number (chemistry)|magic numbers]].<ref>{{citation|title=Magic numbers in polygonal and polyhedral clusters|first1=Boon K.|last1=Teo|first2=N. J. A.|last2=Sloane|author2-link=Neil Sloane|journal=Inorganic Chemistry|year=1985|volume=24|issue=26|pages=4545–4558|doi=10.1021/ic00220a025|url=http://www2.research.att.com/~njas/doc/magic1/magic1.pdf|access-date=2011-04-08|archive-url=https://web.archive.org/web/20120313220128/http://www2.research.att.com/~njas/doc/magic1/magic1.pdf|archive-date=2012-03-13|url-status=dead}}.</ref><ref name="nano">{{citation|title=Metal nanoparticles: synthesis, characterization, and applications|first1=Daniel L.|last1=Feldheim|first2=Colby A.|last2=Foss|publisher=CRC Press|year=2002|isbn=978-0-8247-0604-3|page=76|url=https://books.google.com/books?id=-u9tVYWfRcMC&pg=PA76}}.</ref>