Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts

A000124 on the OEIS · family polynomial · also known as Lazy caterer's sequence

Weight–level plate of Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts — Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.

Open in the 3-D viewer A000124 on the OEIS

Terms	100,000 (n = 0 … 99,999)
Decomposable (a > 2d)	99,996
Level class, k > L	99,996 · 100.00 %
Weight class, k ≤ L	0 · 0.00 %
Ties, k = L	0
On the level line L = 1	1,342
Forced level, l ≤ d²	99,996
Range of a(n)	1 … 4,999,950,001
Range of the jump d	1 … 100,000
Largest weight k, level L	4,987,757,503, 49,770

n(n + 1)/2 + 1, the triangular numbers moved up by one. With c = 1/2 > 1/4 the terms are all forced level (100 %), like the triangular numbers, but the line L = 1 is much thinner: 1,342 terms against 2,642.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.