The numbers in the second column of Pascal's Triangle (1, 3, 6, 10,
15, 21, ...) are called the triangular numbers because they count
the number of objects it takes to form a triangle in the same way the square
numbers count the number of objects needed to form a square. Let's look
at the first few examples:
In general, the numbers of dots needed to make a particular
figure (square, cube, triangle, pyramid, etc.) are called the figurate
numbers. It's easy enough to see that if we take a triangle and
the triangle that is one size smaller and turn the smaller one around, we
can make a square:
The general pattern we can use for the nth
and (n-1)th triangular numbers is as follows:
n
+ (n-1) + (n-2)
+ ...+ 2 +
1 +
1 +
2 + ...+ (n-2)
+ (n-1) = n
+ n +
n +
...+ n
+ n
= n2
From this we can see the pattern we detected early continues forever.
There is also a pattern in the nth column of Pascal's Triangle which
gives us the nth powers of the integers for every positive n, which
is a little more complicated but can be seen at this link.
