decompwlj 3D

Expansion of g.f. x/((1 - x)^2*(1 - x^3))

A001840 on the OEIS · family polynomial

Weight–level plate of Expansion of g.f. x/((1 - x)^2*(1 - x^3))
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 viewerA001840 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L65,734 · 65.74 %
Weight class, k ≤ L34,262 · 34.26 %
Ties, k = L10
On the level line L = 15
Forced level, l ≤ d²5
Range of a(n)0 … 1,666,683,333
Range of the jump d1 … 33,334
Largest weight k, level L99,991, 49,769

33,334 different gaps occur, from 1 to 33,334; the level share is 65.74 %.

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.