decompwlj 3D

Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal lattice)

A003215 on the OEIS · family polynomial · also known as Centered hexagonal numbers

Weight–level plate of Hex (or centered hexagonal) numbers: 3*n*(n+1)+1 (crystal ball sequence for hexagonal 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 viewerA003215 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 = 113,789
Forced level, l ≤ d²99,996
Range of a(n)1 … 29,999,700,001
Range of the jump d6 … 600,000
Largest weight k, level L29,992,500,463, 49,531

3n(n + 1) + 1, the hex numbers. d = 6(n + 1) and a ~ 3 n^2 < d^2: forced level at every decomposable term (100 %). L = 1 holds 13.8 % of the class, then L = 5 (6.9 %).

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.