decompwlj 3D

a(n) = (8*n^3 + 3*n^2 + n) / 6

A219054 on the OEIS · family polynomial

Weight–level plate of a(n) = (8*n^3 + 3*n^2 + n) / 6
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 viewerA219054 on the OEIS
Terms100,000 (n = 1 … 100,000)
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 = 17,793
Forced level, l ≤ d²99,994
Range of a(n)2 … 1,333,338,333,350,000
Range of the jump d11 … 40,000,500,002
Largest weight k, level L1,332,858,388,947,853, 33,231

Every gap is different, from 11 to 40,000,500,002; 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.