decompwlj 3D

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

A001950 on the OEIS · family Beatty · also known as Upper Wythoff sequence

Weight–level plate of Upper Wythoff sequence (a Beatty sequence): a(n) = floor(n*phi^2), where phi = (1+sqrt(5))/2
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 viewerA001950 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L15,381 · 15.38 %
Weight class, k ≤ L84,618 · 84.62 %
Ties, k = L41
On the level line L = 18,752
Forced level, l ≤ d²3
Range of a(n)2 … 261,803
Range of the jump d2 … 3
Largest weight k, level L261,721, 87,261

floor(n phi^2) = floor(n phi) + n, the complement of the lower Wythoff sequence. The gaps are 2 and 3 only (38 % and 62 %). The level share, 15.38 %, is above the lower sequence's 12.20 %; the level class lies on the lines L = 1, 2 and 3.

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.