decompwlj 3D

a(0) = 1; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1)))

A002984 on the OEIS · family self-referential

Weight–level plate of a(0) = 1; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1)))
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 viewerA002984 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L99,997 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 14,549
Forced level, l ≤ d²50,005
Range of a(n)1 … 2,499,265,600
Range of the jump d1 … 49,992
Largest weight k, level L2,498,865,679, 49,788

49,992 different gaps occur, from 1 to 49,992; the level share is 100.00 %; 50.0 % of terms are forced level (l <= d^2); there are no ties.

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.