decompwlj 3D

a(0)=1; a(n) = a(n-1) + lead(a(n-1)) for n > 0 where for an integer x lead(x) is the leading digit in base 10

A061681 on the OEIS · family self-referential

Weight–level plate of a(0)=1; a(n) = a(n-1) + lead(a(n-1)) for n > 0 where for an integer x lead(x) is the leading digit in base 10
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 viewerA061681 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L10,351 · 10.35 %
Weight class, k ≤ L89,645 · 89.65 %
Ties, k = L31
On the level line L = 17,456
Forced level, l ≤ d²0
Range of a(n)1 … 168,587
Range of the jump d1 … 9
Largest weight k, level L168,559, 84,293

9 different gaps occur, from 1 to 9; the level share is 10.35 %; L = 1 holds 72 % 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.