decompwlj 3D

List of binary palindromes of even length (written in base 10)

A048701 on the OEIS · family binary rule

Weight–level plate of List of binary palindromes of even length (written 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 viewerA048701 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,982
Level class, k > L87,816 · 87.83 %
Weight class, k ≤ L12,166 · 12.17 %
Ties, k = L0
On the level line L = 11
Forced level, l ≤ d²54,095
Range of a(n)0 … 13,107,196,611
Range of the jump d3 … 4,294,967,298
Largest weight k, level L4,368,582,209, 140,053,281

32 different gaps occur, from 3 to 4,294,967,298; the level share is 87.83 %; 54.1 % of terms are forced level (l <= d^2); there are no ties; 18 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.