decompwlj 3D

Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1

A000125 on the OEIS · family polynomial · also known as Cake numbers

Weight–level plate of Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1
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 viewerA000125 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,993
Level class, k > L99,993 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 1221
Forced level, l ≤ d²99,993
Range of a(n)1 … 166,661,666,800,000
Range of the jump d1 … 4,999,950,001
Largest weight k, level L161,706,699,181,499, 33,196

(n^3 + 5n + 6)/6, the most pieces from n plane cuts of a cube. A cubic sequence with d = T(n) + 1. Every decomposable term is forced level (l <= d^2).

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.