decompwlj 3D

a(n) = 2*n^2

A001105 on the OEIS · family polynomial · also known as Twice squares

Weight–level plate of a(n) = 2*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 viewerA001105 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 = 12
Forced level, l ≤ d²99,995
Range of a(n)0 … 19,999,600,002
Range of the jump d2 … 399,998
Largest weight k, level L9,997,400,167, 49,727

2n^2. Every term and every gap is even, so 2 | l, and the line L = 1 holds only 2 terms. Every decomposable term is forced level (l <= d^2).

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.