The rascal triangle is a number triangle with numbers arranged in staggered rows, indexed by 
and 
, such that
 |
(1)
|
Anggoro et al. (2010) studied the rascal triangle. It is cataloged in the Online Encyclopedia of Integer Sequences (OEIS A077028)
as Triangle with Diagonal 
Congruent to 1 Mod 
. The congruence is interpreted
for 
.
One common point of exposition among literature regarding the rascal triangle is its similarity to Pascal's triangle. Indeed,
the rascal triangle is topically similar to Pascal's
triangle in that the configuration starting with 
begins
 |
(2)
|
and that the rows afterwards have their first and last entries equal to
 |
(3)
|
respectively.
The similarities don't end there, however. One well-known fact about Pascal's triangle is that the interior entries of each row are determined by the so-called inverted triangle formula
 |
(4)
|
as shown in the diagram above.
On the other hand, the rows of the rascal triangle have interior entries which are determined by a so-called diamond formula of the form
 |
(5)
|
whereby one can compute the first few rows of the rascal triangle to be of the form
 |
(6)
|
(OEIS A077028). Surprisingly, each interior entry of the rascal triangle remains integer-valued despite the fact that, as shown in (5), every such entry is computed using integer division (Anggoro et al. 2010).
By examining the first few rows of the rascal triangle, one can hypothesize the validity of a number of significant number-theoretic properties thereof. For example, the
values given support the congruence claim (OEIS A077028) that each element
of the 
th
diagonal is congruent to 1 modulo 
, e.g., in the fourth diagonal, where each of 1, 4, 7,
10, ... is congruent to 1 modulo 3. Here, 
and the first diagonal consists entirely of 1s. Moreover,
the entry 
lies on the 
th
diagonal when diagonals are numbered starting with 1 (Anggoro et al. 2010).
This observation is fundamental in proving that the rascal triangle consists strictly
of integers.
There are a number of other colorful properties of the rascal triangle. For example, the sums of the rows of the rascal triangle are the so-called cake numbers (OEIS A000125):
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
Several other authors have noted interesting connections between numbers in the rascal triangle and various other fields of mathematics including group theory, linear algebra, and geometric analysis.
