Hex Number
المؤلف:
Conway, J. H. and Guy, R. K
المصدر:
The Book of Numbers. New York: Springer-Verlag
الجزء والصفحة:
...
18-12-2020
1104
Hex Number
A hex number, also called a centered hexagonal number, is given by
where 
is the 
th triangular number and the indexing with 
is used following Conway and Guy (1996). The first few hex numbers for 
, 1, ... are 1, 7, 19, 37, 61, 91, 127, 169, ... (OEIS A003215).
The hex numbers satisfy the recurrence equation
 |
(3)
|
The generating function of the hex numbers is
 |
(4)
|
The hex numbers are related to the cubic numbers by
 |
(5)
|
This follows immediately from the fact that 
, giving a telescoping sum.
The first triangular hex numbers are 1, 91, 8911, 873181, 85562821, ... (OEIS A006244). These correspond to indices 
of triangular and hex numbers 
of 
, 5, 54, 539, 5340, 52865, 523314, 5180279, 51279480, ... (OEIS A087125) and 
, 13, 133, 1321, 13081, 129493, 1281853, ... (OEIS A031138). These are given by solving the Diophantine equation
 |
(6)
|
The first few square hex numbers are 1, 169, 32761, 6355441, ... (OEIS A006051). These correspond to indices 
of triangular and hex numbers 
of 
, 7, 104, 1455, 20272, 282359, 3932760, ... (OEIS A001921) and 
, 13, 181, 2521, 35113, 489061, 6811741, ... (OEIS A001570). These are given by solving the Diophantine equation
 |
(7)
|
The only hex number that is both square and triangular is 1.
There are no cubic hex numbers.
The prime hex numbers are sometimes known as Cuban primes.
REFERENCES:
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 41, 1996.
Gardner, M. "Hexes and Stars." Ch. 2 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 15-25, 1988.
Hindin, H. "Stars, Hexes, Triangular Numbers, and Pythagorean Triples." J. Recr. Math. 16, 191-193, 1983-1984.
Sloane, N. J. A. Sequences A001570/M4915, A001921/M4455, A003215/M4362, A006051/M5409, A006244/M5363, A031138 and A087125 in "The On-Line Encyclopedia of Integer Sequences."
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