A hex number, also called a centered hexagonal number, is given by
(1)
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(2)
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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)
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The generating function of the hex numbers is
(4)
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The hex numbers are related to the cubic numbers by
(5)
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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)
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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)
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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.