decompwlj 3D

Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ..

A001082 on the OEIS · family polynomial · also known as Generalized octagonal numbers

Weight–level plate of Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ..
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 viewerA001082 on the OEIS
Terms100,000 (n = 1 … 100,000)
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 = 15,186
Forced level, l ≤ d²99,995
Range of a(n)0 … 7,499,900,000
Range of the jump d1 … 200,000
Largest weight k, level L7,498,800,047, 73,931

m(3m - 2) for m = 0, 1, -1, 2, -2, ... Two quadratics interleaved: the gaps alternate 4j and 2j + 1 while a ~ 3j^2, so both leave l <= d^2. 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.