Lazy caterer's sequence

As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ≥ 2.

The sequence can be alternatively derived from the sum of up to the first 3 terms of each row of Pascal's triangle:[1]

The difference between successive cake numbers gives the lazy caterer's sequence.

A cut line can always cross over all previous cut lines, as rotating the knife at a small angle around a point that is not an existing intersection will, if the angle is small enough, intersect all the previous lines including the last one added.

Thus, the total number of pieces after n cuts is This recurrence relation can be solved.

Pancake cut into seven pieces with three straight cuts.
The maximum number of pieces, p obtainable with n straight cuts is the n -th triangular number plus one, forming the lazy caterer's sequence (OEIS A000124)
The lazy caterer's sequence (green) and other OEIS sequences in Bernoulli's triangle
Proof without words that summing up to the first 3 terms on each row of Pascal's triangle is equivalent to summing up to the first 2 odd terms of the next row
The maximum number of pieces from consecutive cuts are the numbers in the Lazy Caterer's Sequence.