decompwlj 3D

The squares: a(n) = n^2

A000290 on the OEIS · family polynomial · also known as Squares

Weight–level plate of The squares: a(n) = n^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 viewerA000290 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,995
Level class, k > L99,995 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 110,122
Forced level, l ≤ d²99,995
Range of a(n)0 … 9,999,800,001
Range of the jump d1 … 199,999
Largest weight k, level L9,997,400,167, 49,727

d = 2n + 1 and l = (n - 1)^2 - 2 < d^2, so every decomposable term is forced level (100 %). For a ~ c n^2 in general l/d^2 -> 1/(4c), so every quadratic with c > 1/4 ends up entirely level (A002620, c = 1/4, sits on the edge); the nine polynomial-type sequences here differ only in their ray structure.

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.