decompwlj 3D

Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z; then sequence gives Z values

A001844 on the OEIS · family polynomial · also known as Centered square numbers

Weight–level plate of Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z; then sequence gives Z values
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 viewerA001844 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L99,996 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 18,412
Forced level, l ≤ d²99,996
Range of a(n)1 … 19,999,800,001
Range of the jump d4 … 400,000
Largest weight k, level L19,997,800,057, 49,609

2n(n + 1) + 1 = n^2 + (n + 1)^2. d = 4(n + 1) and a ~ 2 n^2 < d^2: forced level at every decomposable term (100 %). Every l is odd and the fullest line is L = 3 (10.9 %), ahead of L = 1 (8.4 %).

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.