decompwlj 3D

Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers

A016754 on the OEIS · family polynomial · also known as Odd squares

Weight–level plate of Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers
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 viewerA016754 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L99,996 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 120,818
Forced level, l ≤ d²99,996
Range of a(n)1 … 39,999,600,001
Range of the jump d8 … 800,000
Largest weight k, level L39,994,800,161, 49,721

(2n+1)^2. d = 8(n+1) and l = (2n+1)^2 - 8(n+1) is odd. 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.