decompwlj 3D

a(n) = floor(n*phi^7), where phi is the golden ratio, A001622

A004922 on the OEIS · family Beatty

Weight–level plate of a(n) = floor(n*phi^7), where phi is the golden ratio, A001622
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 viewerA004922 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L32,357 · 32.36 %
Weight class, k ≤ L67,640 · 67.64 %
Ties, k = L8
On the level line L = 17,294
Forced level, l ≤ d²27
Range of a(n)0 … 2,903,415
Range of the jump d29 … 30
Largest weight k, level L2,902,397, 96,765

The gaps are 29 and 30; the level share is 32.36 %.

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.