Simply put, triangular squares are triangle numbers that are also squares. For example, T8 = ½· 8 · 9 = 36 = 62. In general a triangular square is any number n, such that n = ½ a(a + 1) and n = b2, where a and b are whole numbers. So,
In
order for b to be a whole number either 
is
a perfect square and
is
a perfect square (for a even), or a is
a perfect square
and
is
a perfect square (for a odd). For example, in the
case of T8
above, 
which
is a perfect square, and a + 1 = 9, which is also
a perfect
square. The next triangular square is T49 = ½ ·
49 · 50 = 1225 = 352. In this case, a =
49
is a perfect square and 
is
a perfect square. Is there a way to generate the number a
so
that we can find any triangular square without having to search to
list of triangle numbers for squares (or the list of squares for
triangle numbers)? The fourth triangular square—the first one
is T1 = 1 = 12—is T288
= ½
· 288 · 289 = 41,616 = 2042. So we have
three series of numbers to work with: the triangle numbers (1, 8, 49,
288, ...), the squares (1, 6, 35, 204, ...), and the triangular
squares themselves (1, 36, 1225, 41616, ...). Of course, it's easy to
find more triangular squares using a computer. (The next one is
1,413,721. See for yourself.) But I would like to find a general
method for generating triangular squares from the natural numbers (1,
2, 3, 4, ...).
As a first step toward our goal, we take note of the fact that the ratio between successive squares is always between 5 and 6. In fact, it appears that 35 = 6 · 6 – 1 and 204 = 6 · 35 – 6. Let's see if 6 · 204 – 35 = 1189 also leads to a solution. To do so, we need to solve the equation given above for a.
For b = 1189, a = 1681. A quick check of squares between 2042 and 11892 (and triangle numbers between T288 and T1681) shows that we didn't miss any in between. This makes a good working method for generating triangular squares, but we are still missing a rigorous proof that the method works for all triangular squares and captures all of them. We also don't have a method for finding the nth triangular square without generating all the 1 through n – 1 triangular squares.