decompwlj 3D

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

A001951 on the OEIS · family Beatty · also known as Beatty sequence floor(n sqrt 2)

Weight–level plate of A Beatty sequence: a(n) = floor(n*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 viewerA001951 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L11,372 · 11.37 %
Weight class, k ≤ L88,625 · 88.63 %
Ties, k = L54
On the level line L = 19,302
Forced level, l ≤ d²1
Range of a(n)0 … 141,419
Range of the jump d1 … 2
Largest weight k, level L141,413, 70,708

floor(n sqrt 2), a Beatty sequence with gaps 1 and 2 only. The level share is 11.37 %, on the lines L = 1 (82 %) and L = 2.

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.