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A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

A (1,2) leaper is a chess knight.

a(2)-a(5) were computed by Daniël Karssen.

This is a complete list of all the possible ending 'trapped' square values for the knight (2 by 1 leaper) starting from any square. The list has 1518 values - the knight starting from any square on the infinite board will eventually be trapped on a square with one of these numbers.

I do not have a proof this is the complete list of all ending values but I believe it is correct. I have checked every knight starting square up to 100000 and they all end on one of these 1518 squares. I then check further out to 110000 and ensure these paths always move inwards once they pass the square of values which contains the 100000 value, and check they do not move outwards again passed this square. As every knight sequence out to infinity would have to cross/land between this 100000 to 110000 group of values (as they are attracted toward square 1 due to their lowest-available-value preference), and as all values have been checked inside these, it implies all knight paths with starting square values out to infinity eventually end on this list of 1518 squares.

Also note this is the ordered sequence of all 1518 squares - the initial value found starting the knight at square 1 is 2084.

Construction: with a knight (a (1,2)-leaper) on an infinite spiral numbered board moving to the lowest numbered unvisited square (see A316884), start the knight on any square and continue moving it until it is trapped. Then start an entirely new sequence starting the knight at the same square at which it was previously trapped. Continue this process until the square at which the knight is trapped has occurred previously, indicating an end square loop. All starting squares for the knight on the infinite board will eventually lead to the knight path falling into one of the 3 end square loops listed here.

As the total number of squares in which the knight can be trapped is finite (see A306291), we expect there to be a finite number of end square loops - in theory, only those values (1518 is all) need to be checked when searching for an end square loop. However, all starting square values up to 302500 have been checked to determine into which of the 3 found loops the knight eventually falls. The 13-member loop with 406 as its lowest number is found to be the dominant loop, with about 89.6% of all initial starting squares going to it. The other 10.4% mostly go to the 3-member loop with 404 as its lowest number, with a decreasingly small remainder going to the 2-member loop with 910 as it lowest number. The attached 3-color image showing the start-value-to-loop mapping shows that the pattern of starting square to end square loops becomes very regular away from the center of the board.

For a knight (a (1,2) leaper) starting at square 1 and moving on a spirally numbered 2D grid to the lowest-numbered available square at each step (see A316667), a(n) is the number of the square at which the knight is trapped if it is allowed to visit each square no more than n times -- the knight is not trapped until each of the 8 surrounding squares to which it can leap has been visited n times. The choice of the square to which it goes at each step is determined solely by the square with the lowest spiral number, as long as it has been visited fewer than n times. This is an infinite sequence, although end squares beyond a(35) are currently unknown.