decompwlj 3D

Tetrahedron-tree numbers: a(n)=sum(b(m),m=1..n), b(m)=1, 1,3, 1,3,6, 1,3,6,10,..., 1,2,...,i*(i+1)2

A051677 on the OEIS · family summatory

Weight–level plate of Tetrahedron-tree numbers: a(n)=sum(b(m),m=1..n), b(m)=1, 1,3, 1,3,6, 1,3,6,10,..., 1,2,...,i*(i+1)2
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 viewerA051677 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L69,419 · 69.42 %
Weight class, k ≤ L30,577 · 30.58 %
Ties, k = L5
On the level line L = 12,734
Forced level, l ≤ d²36,258
Range of a(n)1 … 1,676,380,656
Range of the jump d1 … 99,681
Largest weight k, level L1,675,788,701, 820,623,832

446 different gaps occur, from 1 to 99,681; the level share is 69.42 %; 36.3 % of terms are forced level (l <= d^2).

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.