decompwlj 3D

a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game

A001954 on the OEIS · family Beatty

Weight–level plate of a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game
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 viewerA001954 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,998
Level class, k > L17,270 · 17.27 %
Weight class, k ≤ L82,728 · 82.73 %
Ties, k = L36
On the level line L = 18,544
Forced level, l ≤ d²3
Range of a(n)1 … 341,419
Range of the jump d3 … 4
Largest weight k, level L341,357, 85,349

The gaps are 3 and 4; the level share is 17.27 %; L = 1 holds 49 % 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.