decompwlj 3D

a(n) = floor(n*phi^4), where phi is the golden ratio, A001622

A004919 on the OEIS · family Beatty

Weight–level plate of a(n) = floor(n*phi^4), 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 viewerA004919 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L22,263 · 22.26 %
Weight class, k ≤ L77,734 · 77.74 %
Ties, k = L18
On the level line L = 18,083
Forced level, l ≤ d²6
Range of a(n)0 … 685,403
Range of the jump d6 … 7
Largest weight k, level L685,231, 97,909

The gaps are 6 and 7; the level share is 22.26 %; L = 1 holds 36 % of the level class.

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.