decompwlj 3D

Palindromes in base 10

A002113 on the OEIS · family digit rule · also known as Base-10 palindromes

Weight–level plate of Palindromes 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 viewerA002113 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L71,652 · 71.66 %
Weight class, k ≤ L28,343 · 28.34 %
Ties, k = L16
On the level line L = 14,699
Forced level, l ≤ d²9,211
Range of a(n)0 … 900,000,009
Range of the jump d1 … 11,000
Largest weight k, level L899,990,009, 37,499,999

Starts at a(1) = 0. Only 13 distinct gaps below 10^12, all of the form 10^j or 11*10^j (plus the single value 2). The plane is self-similar with period 2 in the digit length; the level share oscillates 60/94 % with its parity.

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.