decompwlj 3D

Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4)

A002620 on the OEIS · family polynomial · also known as Quarter-squares

Weight–level plate of Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4)
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 viewerA002620 on the OEIS
Terms100,000 (n = 1 … 100,000)
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 = 1707
Forced level, l ≤ d²99,995
Range of a(n)0 … 2,500,000,000
Range of the jump d1 … 50,000
Largest weight k, level L2,499,200,063, 49,728

floor(n^2/4), from a(1) = 0 (a(0) = a(1) = 0). The gap takes each value twice and l <= d^2 throughout: forced level at every decomposable term (100 %).

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.