decompwlj 3D

Triangular numbers together with squares (excluding 0)

A005214 on the OEIS · family polynomial · also known as Triangular numbers and squares

Weight–level plate of Triangular numbers together with squares (excluding 0)
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 viewerA005214 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L83,776 · 83.78 %
Weight class, k ≤ L16,220 · 16.22 %
Ties, k = L0
On the level line L = 16,183
Forced level, l ≤ d²41,426
Range of a(n)1 … 1,716,013,236
Range of the jump d1 … 58,583
Largest weight k, level L1,715,940,899, 130,185,247

The union of two sequences that are 100 % level-classified is not: merging triangular numbers and squares shrinks the gaps below sqrt(l) on 58.6 % of terms, and 16.2 % turn weight. No ties occur.

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.