I recently stumbled on a formula for the largest gap between consecutive items in a row of Pascal’s triangle.
For n ≥ 2, where 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.