Almost diagonals in Pascal’s triangleProperties
The first and most obvious property of Pascal’s triangle is its symmetry. If we
draw a vertical line down through the middle, the triangle has ‘mirror symmetry’
- it is the same to the left of the vertical line as to the right of it. This allows us
to talk about plain ‘diagonals’, because a northeast diagonal will be the same as a
northwest diagonal. Under the diagonal made up of 1s we have the diagonal
made up of the counting numbers 1, 2, 3, 4, 5, 6,... Under that there are the
triangular numbers, 1, 3, 6, 10, 15, 21,... (the numbers which can be made up
of dots in the form of triangles). In the diagonal under that we have the
tetrahedral numbers, 1, 4, 10, 20, 35, 56,... These numbers correspond to
tetrahedra (‘three-dimensional triangles’, or, if you like, the number of cannon
balls which can be placed on triangular bases of increasing sizes). And what
about the ‘almost diagonals’?
If we add up the numbers in lines across the triangle (which are not rows or
true diagonals), we get the sequence 1, 2, 5, 13, 34,... Each number is three
times the previous one with the one before that subtracted. For example 34 = 3
× 13 – 5. Based on this, the next number in the sequence will be 3 × 34 – 13 =
- We have missed out the alternate ‘almost diagonals’, starting with 1, 1 + 2 =
3, but these will give us the sequence 1 , 3 , 8 , 21 , 55 ,... and these are
generated by the same ‘3 times minus 1’ rule. We can therefore generate the next
number in the sequence, as 3 × 55 – 21 = 144. But there’s more. If we
interleave these two sequences of ‘almost diagonals’ we get the Fibonacci