decompwlj 3D

Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n

A000217 on the OEIS · family polynomial · also known as Triangular numbers

Weight–level plate of Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n
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 viewerA000217 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,995
Level class, k > L99,995 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 12,642
Forced level, l ≤ d²99,995
Range of a(n)0 … 4,999,950,000
Range of the jump d1 … 100,000
Largest weight k, level L4,998,750,077, 49,769

d = n + 1 and l = (n + 1)(n - 2)/2 = d(n - 2)/2, so l <= d^2 always and L/k < 1/2: every term is level-classified and the cloud stops a factor of two short of the diagonal.

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.