decompwlj 3D

a(n) = n*(2*n^2 + n + 1)/2

A085786 on the OEIS · family polynomial

Weight–level plate of a(n) = n*(2*n^2 + n + 1)/2
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 viewerA085786 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 = 111,221
Forced level, l ≤ d²99,994
Range of a(n)2 … 1,000,005,000,050,000
Range of the jump d9 … 30,000,400,002
Largest weight k, level L999,915,001,849,987, 33,097

Every gap is different, from 9 to 30,000,400,002; 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.