decompwlj 3D

Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + n-k)/2, n, k > 0, read by antidiagonals

A014132 on the OEIS · family complement · also known as Non-triangular numbers

Weight–level plate of Complement of triangular numbers (A000217); also array T(n,k) = ((n+k)^2 + n-k)/2, n, k > 0, read by antidiagonals
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 viewerA014132 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L9,514 · 9.51 %
Weight class, k ≤ L90,485 · 90.49 %
Ties, k = L62
On the level line L = 19,514
Forced level, l ≤ d²1
Range of a(n)2 … 100,447
Range of the jump d1 … 2
Largest weight k, level L100,417, 50,223

Not a triangular number. Gaps 1, and 2 across each triangular number. The level share is 9.51 %; L = 1 holds 100 % of the level class.

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.