decompwlj 3D

Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order

A007931 on the OEIS · family digit rule

Weight–level plate of Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order
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 viewerA007931 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,969
Level class, k > L11,086 · 11.09 %
Weight class, k ≤ L88,883 · 88.91 %
Ties, k = L0
On the level line L = 14,136
Forced level, l ≤ d²613
Range of a(n)1 … 2,111,122,121,211,112
Range of the jump d1 … 888,888,888,888,889
Largest weight k, level L2,111,122,121,111,033, 1,055,561,060,605,555

16 different gaps occur, from 1 to 888,888,888,888,889; the level share is 11.09 %; L = 1 holds 37 % of the level class; there are no ties; 31 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.