The square-triangle theorem states that any nonnegative integer can be represented as the sum of a square, an even square, and a triangular number (Sun 2005), i.e.,
 |
(1)
|
for 
,
,
and 
integers. For example,
 |  |  |
(2)
|
 |  |  |
(3)
|
corresponding to the solutions 
and (3,1,6), respectively.
Values of 
lacking a representation in which all of 
, 
, and 
all are nonzero are 1, 2, 3, 4, 7, 10, 12, 22, and 24 (OEIS
A118426).
The following table gives solutions for the first few 
.
 | solutions  |
| 1 |  ,
 ,
 ,
 ,
 ,
 |
| 2 |  ,  ,  ,  |
| 3 |  ,
 |
| 4 |  ,  ,  ,  ,  ,  ,  ,  ,  ,  |
| 5 |  ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 |
| 6 |  ,  ,  ,  ,  ,  ,  ,  ,  ,  |
| 7 |  ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 |
| 8 |  ,  ,  ,  ,  ,  ,  ,  ,  ,  |
| 9 |  ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 ,
 |
| 10 |  ,  ,  ,  ,  ,  ,  ,  ,  ,  |
The numbers of solutions for 
, 2, ... are 6, 4, 2, 12, 16, 10, 12, 16, 12, 14, 20, 4,
8, 24, 14, ... (OEIS A118421). The high-water
marks are 6, 12, 16, 20, 24, 28, 32, 40, 44, 56, 60, 72, 80, 88, 96, 108, ... (OEIS
A118422), which occur for 
, 4, 5, 11, 14, 19, 20, 23, 26, 41, 53, 68, 86, 110, 145,
... (OEIS A118423).
."
9 May 2005.