I recently stumbled on a formula for the largest gap between consecutive items in a row of Pascal’s triangle.
For n ≥ 2,
For example, consider the 6th row of Pascal’s triangle, the coefficients of (x + y)6.
1, 6, 15, 20, 15, 6, 1
The largest gap is 9, the gap between 6 and 15 on either side. In our formula n = 6 and so
τ = (8 – √8)/2 = 2.5858
and so the floor of τ is 2. The equation above says the maximum gap should be between the binomial coefficients with k = 2 and 1, i.e. between 15 and 6, as we expected.
I’ve needed a result like this in the past, but I cannot remember now why. I’m posting it here for my future reference and for the reference of anyone else who might need this. I intend to update this post if I run across an application.
More on Pascal’s triangle
- Pascal’s triangle and Fermat’s little theorem
- Distribution of numbers in Pascal’s triangle
- The Star of David theorem
Source: Zun Shan and Edward T. H. Wang. The Gaps Between Consecutive Binomial Coefficients. Mathematics Magazine, Vol. 63, No. 2 (Apr., 1990), pp. 122–124