decompwlj 3D

Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2 = A001622

A000201 on the OEIS · family Beatty · also known as Lower Wythoff sequence

Weight–level plate of Lower Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi), where phi = (1+sqrt(5))/2 = 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 viewerA000201 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L12,196 · 12.20 %
Weight class, k ≤ L87,802 · 87.80 %
Ties, k = L44
On the level line L = 19,164
Forced level, l ≤ d²1
Range of a(n)1 … 161,803
Range of the jump d1 … 2
Largest weight k, level L161,783, 80,897

floor(n phi), a Beatty sequence. The gaps are 1 and 2 only, in the Fibonacci-word pattern, so k is the least divisor of a - d above 1 or 2, as for the naturals and the odd numbers. The level share, 12.20 %, falls between theirs.

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.