decompwlj 3D

Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a

A191113 on the OEIS · family self-referential

Weight–level plate of Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a
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 viewerA191113 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L21,918 · 21.92 %
Weight class, k ≤ L78,078 · 78.08 %
Ties, k = L0
On the level line L = 13
Forced level, l ≤ d²2,814
Range of a(n)1 … 540,791,104
Range of the jump d1 … 14,374,958
Largest weight k, level L270,395,513, 180,263,700

850 different gaps occur, from 1 to 14,374,958; the level share is 21.92 %; 2.8 % of terms are forced level (l <= d^2); L = 2 holds 38 % of the level class; there are no ties.

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.