decompwlj 3D

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

A117560 on the OEIS · family polynomial

Weight–level plate of a(n) = n*(n^2 - 1)/2 - 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 viewerA117560 on the OEIS
Terms100,000 (n = 2 … 100,001)
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 = 114,015
Forced level, l ≤ d²99,994
Range of a(n)2 … 500,015,000,099,999
Range of the jump d9 … 15,000,450,003
Largest weight k, level L499,925,003,399,951, 32,939

Every gap is different, from 9 to 15,000,450,003; 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.