decompwlj 3D

Centered octahedral numbers (crystal ball sequence for cubic lattice)

A001845 on the OEIS · family polynomial · also known as Centered octahedral numbers

Weight–level plate of Centered octahedral numbers (crystal ball sequence for cubic lattice)
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 viewerA001845 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,993
Level class, k > L99,993 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 114,690
Forced level, l ≤ d²99,993
Range of a(n)1 … 1,333,313,333,599,999
Range of the jump d6 … 40,000,000,002
Largest weight k, level L1,332,793,405,596,797, 33,067

(2n+1)(2n^2+2n+3)/3, the centered octahedral numbers. A cubic sequence. 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.