Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ...
Open in the 3-D viewerA001318 on the OEIS
| Terms | 100,000 (n = 0 … 99,999) |
|---|---|
| Decomposable (a > 2d) | 99,996 |
| Level class, k > L | 77,602 · 77.61 % |
| Weight class, k ≤ L | 22,394 · 22.39 % |
| Ties, k = L | 3 |
| On the level line L = 1 | 6,729 |
| Forced level, l ≤ d² | 49,999 |
| Range of a(n) | 0 … 3,749,975,000 |
| Range of the jump d | 1 … 99,999 |
| Largest weight k, level L | 3,749,325,029, 74,563 |
m(3m - 1)/2 for m = 0, 1, -1, 2, -2, ... Two quadratics interleaved: the gaps alternate j and 2j + 1. With a ~ 1.5 j^2 the gap j leaves l > d^2 while the gap 2j + 1 forces level, so exactly half the terms (49,999) are forced level. The other half is split, and the level share ends at 77.61 %.
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.