Rhombic Dodecahedral Number
A figurate number which is constructed as a centered cube with a square pyramid appended to each face,
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a centered
cube number and 
is a square pyramidal number. The first few are
1, 15, 65, 175, 369, 671, ... (OEIS A005917).
The generating function of the rhombic dodecahedral
numbers is
 |
(3)
|
 |  |
A related set of numbers is the number of cubes in the Haűy construction of the rhombic dodecahedron, given by
 |
(4)
|
for 
an odd number.
Re-indexing with 
then gives
 |
(5)
|
giving the first few values 1, 33, 185, 553, 1233, ... (OEIS A046142).
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