decompwlj 3D

Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+sqrt(2)))

A003151 on the OEIS · family Beatty · also known as floor(n(1 + sqrt 2))

Weight–level plate of Beatty sequence for 1+sqrt(2); a(n) = floor(n*(1+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 viewerA003151 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L14,919 · 14.92 %
Weight class, k ≤ L85,079 · 85.08 %
Ties, k = L44
On the level line L = 18,904
Forced level, l ≤ d²1
Range of a(n)2 … 241,421
Range of the jump d2 … 3
Largest weight k, level L241,391, 80,473

floor(n(1 + sqrt 2)), the complement of floor(n(1 + 1/sqrt 2)). Gaps 2 and 3. The level share is 14.92 %; L = 1 holds 60 % 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.