Patterns from the Diagonals of Pascal’s Triangle

Common sequences which are discussed in Pascal's Triangle include the counting numbers and triangle numbers from the diagonals of Pascal's Triangle. By examining these diagonals, however, not only do we find these two sequences, but a whole shower of sequences, which appear to get ever more complicated, each one a development of the last one. In this article, I discuss these sequences and how they are linked to each other.

Sequences can be found in the diagonals of Pascal's triangle. In the next diagram, however, as it is left justified, we need to carefully examine the columns, and see what sequences we can spot.

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Working our way inwards, in the first column, the numbers are always 1. Next, we have the counting numbers in order in the second column. Then, we have the slightly more complicated sequence of the Triangle numbers (These are sums of all previous counting numbers - for example, the 8th triangle number is 1+2+3+4+5+6+7+8 = 36. Also, by creating triangles of dots or counters, and counting the number of these dots used, you can get the triangle numbers) If you take the differences of consecutive triangle numbers, you get the counting numbers. This is shown below:

Sequence: 1 3 6 10 15 21 28

Differences: 2 3 4 5 6 7

The sequence in the fourth column is more complicated again. The numbers 1,4,10,20,35 are called tetrahedral numbers. Like triangle numbers, these can be understood visually. This time, you need to create triangular based pyramids (NOT square based pyramids like those in Egypt). For your pyramids, you could use coins, counters, marbles - whatever you like. Like before, all you have to do is count the number of things used to create your pyramid. Also, we can also express the tetrahedral numbers in a "difference tree".

1 4 10 20 35 56

3 6 10 15 21

We can begin to see an interesting pattern emerging. The differences of the counting numbers 1,2,3,4,5,6,7,8... from column two of Pascal's triangle are always one, and the first column of Pascal's triangle is also always 1s. Similarly, the triangle numbers 1,3,6,10,15,21... are from column 3 and give differences of 1,2,3,4,5,6... , which is column two, and the tetrahedral numbers from the fourth row (1,4,10,20,35... ) have differences of the triangle numbers from the third row of the triangle.

In fact, this pattern always continues. The differences of one column gives the numbers from the previous column (the first number 1 is knocked off, however). So, for example, if we look at the fifth column in Pascal's triangle, we get the sequence 1,5,15,35,70,126... These are called the pentatope numbers, and appear to be a very complicated sequence. However, their differences just give the tetrahedral numbers, (starting from 4).

It is clear, therefore, that Pascal's triangle is a powerful tool in making sense of these complicated sequences, and contains patterns in its diagonals which are far more extensive than one might intitially imagine.

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