decompwlj 3D

Numbers of the form 3*(x^2 + xy + y^2 + x + y) + 1 where x and y are integers

A202822 on the OEIS · family quadratic form

Weight–level plate of Numbers of the form 3*(x^2 + xy + y^2 + x + y) + 1 where x and y are integers
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 viewerA202822 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L28,379 · 28.38 %
Weight class, k ≤ L71,618 · 71.62 %
Ties, k = L135
On the level line L = 116,013
Forced level, l ≤ d²5
Range of a(n)1 … 844,279
Range of the jump d3 … 57
Largest weight k, level L844,201, 168,830

19 different gaps occur, from 3 to 57; the level share is 28.38 %; L = 1 holds 56 % of the level class.

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.