decompwlj 3D

Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton

A147562 on the OEIS · family self-referential

Weight–level plate of Number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton
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 viewerA147562 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,995
Level class, k > L70,865 · 70.87 %
Weight class, k ≤ L29,130 · 29.13 %
Ties, k = L4
On the level line L = 18,895
Forced level, l ≤ d²26,941
Range of a(n)0 … 10,045,632,969
Range of the jump d1 … 57,395,628
Largest weight k, level L5,726,623,057, 770,891,565

17 different gaps occur, from 1 to 57,395,628; the level share is 70.87 %; 26.9 % of terms are forced level (l <= d^2).

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.