decompwlj 3D

Double-bitters: only even length runs in binary expansion

A001196 on the OEIS · family binary rule

Weight–level plate of Double-bitters: only even length runs in binary expansion
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 viewerA001196 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,983
Level class, k > L14,011 · 14.01 %
Weight class, k ≤ L85,972 · 85.99 %
Ties, k = L0
On the level line L = 11
Forced level, l ≤ d²629
Range of a(n)0 … 16,110,109,695
Range of the jump d3 … 8,589,934,593
Largest weight k, level L5,370,036,547, 3,222,021,897

17 different gaps occur, from 3 to 8,589,934,593; the level share is 14.01 %; L = 3 holds 37 % of the level class; there are no ties; 17 terms do not decompose.

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.