The sequence is for squares at exactly that distance, not squares reachable in that distance or less. Thus, it should increase as order n, not n^2. On Fri, Mar 28, 2014 at 11:20 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Fri, Mar 28, 2014 at 1:16 PM, Marc LeBrun <mlb@well.com> wrote:
="Warren D Smith" <warren.wds@gmail.com> What is the distance between (x1,y1) and (x2,y2) measured in number of knight-moves?
Warren, some years ago I put some "knight metric" sequences in the OEIS. Special-cases to compare with your conjecture: dx,dy =...
...n,0: http://oeis.org/A018837 2[ (n+2)/4 ] if n even, 2[ (n+1)/4 ]+1 if n odd (n >= 8)
...n,n: http://oeis.org/A018839 2*ceiling(n/3), n >= 3
When I follow that link, I don't get knight distance to (n,n); I get a series purporting to be the series for
Squares on infinite chess-board at n moves from center using a {2,3} fairy knight.
But this sequence is incorrect; it claims that the answer is For n >= 8, a(n) = 68n - 72. But that can't be right; you can reach any square within distance n by first going to an appropriate square in a (2,6) neighborhood of the initial square, and then zigzagging to the right to get to a square from which you can get to the desired square in a straight line. Each part of this procedure takes a number of steps linear in n, so the total number of squares at n moves from the center must be of order n^2.
Andy
Also perhaps of interest is the size of the n-move neighborhood, etc. The OEIS has some references as well. Search on "knight" gives many hits.
"Enjoy"! --MLB
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