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A328928 Squares visited by a knight moving on a taxicab geometry numbered board where the knight moves to the smallest numbered unvisited square; the minimum distance from the origin is used if the square numbers are equal; the smallest spiral number ordering is used if the distances are equal.

%I A328928 #26 Mar 31 2024 17:09:29
%S A328928 0,3,2,1,2,1,2,1,2,1,2,3,4,5,2,3,4,3,4,3,2,5,4,3,4,5,2,3,4,5,4,5,4,3,
%T A328928 4,5,6,3,4,3,4,3,4,3,4,5,6,5,6,7,6,5,4,5,4,7,6,5,6,7,6,5,6,5,4,7,6,5,
%U A328928 6,7,6,5,8,7,6,7,6,5,6,7,6,5,6,5,6,7,8,7,8,7,6,7,8,7,6,7,8,7,6
%N A328928 Squares visited by a knight moving on a taxicab geometry numbered board where the knight moves to the smallest numbered unvisited square; the minimum distance from the origin is used if the square numbers are equal; the smallest spiral number ordering is used if the distances are equal.
%C A328928 This sequence uses the taxicab geometry distance from the 0-squared origin to enumerate each square on the board. At each step the knight goes to an unvisited square with the smallest square number. If the knight has a choice of two or more squares with the same number it then chooses the square which is the closest to the 0-squared origin. If two or more squares are found which also have the same distance to the origin, then the square which was first drawn in a square spiral numbering is chosen, i.e., the smallest spiral numbered square as in A316667.
%C A328928 The sequence is finite. After 1092366 steps a square with the number 728 (standard spiral number = 1165673) is visited, after which all neighboring squares have been visited.
%C A328928 See A328908(n) for the position on the spiral (cf. A316328) of the square visited at move n. - _M. F. Hasler_, Nov 04 2019
%H A328928 Scott R. Shannon, <a href="/A328928/b328928.txt">Table of n, a(n) for n = 0..20000</a>
%H A328928 M. F. Hasler, <a href="/wiki/Knight_tours">Knight tours</a>, OEIS wiki, Nov. 2019
%H A328928 Scott R. Shannon, <a href="/A328928/a328928.png">Image showing the 1092366 steps of the knight's path</a>. The green dot is the first square and the red dot the last. Blue dots show the eight occupied squares surrounding the final square; the final square is on the boundary at about the 12:30 clock position.
%H A328928 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).
%H A328928 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%F A328928 a(n) = |A174344(p)| + |A274923(p)| with p = A328908(n)+1. - _M. F. Hasler_, Nov 04 2019
%e A328928 The squares are labeled using their taxicab geometry distance from the origin:
%e A328928 .
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  6 |  5 |  4 |  3 |  4 |  5 |  6 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  5 |  4 |  3 |  2 |  3 |  4 |  5 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  4 |  3 |  2 |  1 |  2 |  3 |  4 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  3 |  2 |  1 |  0 |  1 |  2 |  3 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  4 |  3 |  2 |  1 |  2 |  3 |  4 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  5 |  4 |  3 |  2 |  3 |  4 |  5 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928     |  6 |  5 |  4 |  3 |  4 |  5 |  6 |
%e A328928     +----+----+----+----+----+----+----+
%e A328928 .
%e A328928 If the knight has a choice of two or more squares with the same number which also have the same distance from the origin, then the square with the minimum square spiral number, as shown in A316667, is chosen.
%o A328928 (PARI) A328928(n)=normlp(coords(A328908(n)),1) \\ with coords() defined e.g. in A296030. - _M. F. Hasler_, Nov 04 2019
%Y A328928 Cf. A328908, A316667, A326922.
%Y A328928 Cf. A174344, A274923, A296030 (coordinates of the n-th point on the spiral).
%K A328928 nonn,fini
%O A328928 0,2
%A A328928 _Scott R. Shannon_, Oct 31 2019