OFFSET
0,8
COMMENTS
The last tiebreaking rule, which chooses the colexicographically largest point, seems never to come into play, except for the very first step.
Backtracking (discarding a candidate move because it inevitably leads to a dead end) is needed for the first time in the 359th step, in which the step to (-1,-8) is discarded because all its neighbors are already visited. Instead the knight moves to (-4,-11). (-1,-8) is also the position closest to the origin of the positions that are never visited. Among the first 1000000 steps, backtracking is needed 125 times. The majority of the discarded steps are steps to boundary points (points with |x|==1 and y<0).
The first time the 3rd differences need to be compared is at the 22nd step. Among the first 1000000 steps, differences of order higher than 3 never need to be compared; 3rd differences are needed 5 times.
LINKS
Pontus von Brömssen, Table of n, a(n) for n = 0..10000
Pontus von Brömssen, Illustration of the first 100 steps.
Pontus von Brömssen, Illustration of the first 20000 steps.
Pontus von Brömssen, Plot of path, using Plot2.
CROSSREFS
KEYWORD
sign
AUTHOR
Pontus von Brömssen, Jul 01 2026
STATUS
approved