decompwlj 3D

Numbers not of form "nearest integer to n*tau", tau = (1+sqrt(5))/2

A007064 on the OEIS · family Beatty

Weight–level plate of Numbers not of form "nearest integer to n*tau", tau = (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 viewerA007064 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L15,428 · 15.43 %
Weight class, k ≤ L84,570 · 84.57 %
Ties, k = L37
On the level line L = 18,783
Forced level, l ≤ d²1
Range of a(n)1 … 261,802
Range of the jump d2 … 3
Largest weight k, level L261,799, 87,265

The gaps are 2 and 3; the level share is 15.43 %; L = 1 holds 57 % 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.