decompwlj 3D

Maximal number of pieces obtained by slicing a torus (or a bagel) with n cuts: (n^3 + 3*n^2 + 8*n)/6 (n > 0)

A003600 on the OEIS · family polynomial

Weight–level plate of Maximal number of pieces obtained by slicing a torus (or a bagel) with n cuts: (n^3 + 3*n^2 + 8*n)/6 (n > 0)
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 viewerA003600 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,994
Level class, k > L99,994 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 15,518
Forced level, l ≤ d²99,994
Range of a(n)1 … 166,666,666,749,999
Range of the jump d1 … 5,000,050,001
Largest weight k, level L166,656,666,849,997, 33,086

Every gap is different, from 1 to 5,000,050,001; every decomposable term is forced level (l <= d^2); 6 terms do not decompose.

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.