Octahedral Number
المؤلف:
Conway, J. H. and Guy, R. K.
المصدر:
The Book of Numbers. New York: Springer-Verlag
الجزء والصفحة:
...
22-12-2020
1399
Octahedral Number
A figurate number which is the sum of two consecutive pyramidal numbers,
 |
(1)
|
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
 |
(2)
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Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers (Dickson 2005, p. 23).
A related set of numbers is the number of cubes in the Haűy construction of the octahedron. Each cross section has area
 |
(3)
|
where 
is an odd number, and adding all cross sections gives
 |
(4)
|
for 
an odd number. Re-indexing so that 
gives
 |
(5)
|
the first few values of which are 1, 7, 25, 63, 129, ... (OEIS A001845). These numbers have the generating function
 |
(6)
|
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."
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