A hexagonal square number is both a hexagonal number 
and a square
number 
for positive integers 
and 
. Its indices satisfy 
, or
 |
(1)
|
Completing the square and rearranging gives
 |
(2)
|
Therefore, defining 
and 
gives the Pell equation
 |
(3)
|
The first few solutions are 
, (17, 12), (99, 70), (577, 408), .... These give
the solutions 
,
(9/2, 6), (25, 35), (289/2, 204), ..., giving the positive
integer solutions (1, 1), (25, 35), (841, 1189), (28561, 40391), ... (OEIS A008844 and A046176).
The corresponding hexagonal square numbers are 1, 1225, 1413721, 1631432881, 1882672131025,
... (OEIS A046177).
Closed-form solutions for integers 
are
 |  |  |
(4)
|
 |  | ![1/4{1+1/2[(3-2sqrt(2))^(2k+1)+(3+2sqrt(2))^(2k+1)]},](/images/equations/HexagonalSquareNumber/Inline16.svg) |
(5)
|
giving the 
th
hexagonal square number as
![HS_k=1/(32)[-2+(17-12sqrt(2))(3-2sqrt(2))^(4k)+(17+12sqrt(2))(3+2sqrt(2))^(4k)].](/images/equations/HexagonalSquareNumber/NumberedEquation4.svg) |
(6)
|
A recurrence relation for 
is given by
 |
(7)
|
for integer 
, with 
. The square root 
in the closed-form solution indexed from 
is 
(M. Carreira, pers. comm., Sep. 11, 2004).
