Clark's triangle is a number triangle created by setting the vertex equal to 0, filling one diagonal with 1s, the other diagonal
with multiples of an integer 
, and filling in the remaining entries by summing the elements
on either side from one row above. The illustration above shows Clark's triangle
for 
(OEIS A090850).
Index the rows by 
and the columns by 
.
Set 
.
For 
,
the boundary entries are
 |  |  |
(1)
|
 |  |  |
(2)
|
and the recurrence relation
 |
(3)
|
determines the interior entries for 
. For 
and 
, the result is given analytically by
 |
(4)
|
where 
is a binomial coefficient, taken as 0 when
or 
(M. Alekseyev, pers. comm., Aug. 10, 2005).
For 
,
the entries 
and 
simplify to
 |  |  |
(5)
|
 |  |  |
(6)
|
for 
and 
,
respectively. These columns contain consecutive cubes
and squares of consecutive triangular
numbers.
The sum of the 
th row for 
is given by
 |
(7)
|
(M. Alekseyev, pers. comm., Aug. 10, 2005).
The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Clark's triangle
with 
.
