decompwlj 3D

A Beatty sequence: a(n) = floor(n*(2 + sqrt(2)))

A001952 on the OEIS · family Beatty · also known as floor(n(2 + sqrt 2))

Weight–level plate of A Beatty sequence: a(n) = floor(n*(2 + sqrt(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 viewerA001952 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L17,239 · 17.24 %
Weight class, k ≤ L82,759 · 82.76 %
Ties, k = L31
On the level line L = 18,560
Forced level, l ≤ d²2
Range of a(n)3 … 341,421
Range of the jump d3 … 4
Largest weight k, level L341,333, 85,352

floor(n(2 + sqrt 2)), the complement of floor(n sqrt 2). Gaps 3 and 4. The level share is 17.24 %; L = 1 holds 50 % 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.