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A343563 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the unvisited square containing the spiral number with the smallest digit sum. In case of a tie it chooses the lowest number.

%I A343563 #11 May 07 2021 09:13:12
%S A343563 1,10,3,30,11,4,13,2,5,20,23,6,21,40,105,202,103,100,141,250,315,190,
%T A343563 251,140,61,14,31,12,15,32,55,130,91,180,301,234,127,52,25,50,121,222,
%U A343563 119,220,117,80,51,124,231,126,53,26,9,22,41,106,203,104,201,102,143,252,321,480,323,400,403
%N A343563 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the unvisited square containing the spiral number with the smallest digit sum. In case of a tie it chooses the lowest number.
%C A343563 This sequences gives the numbers of the squares visited by a knight moving on a square-spiral numbered board where at each step the knight moves to the unvisited neighbor one knight-leap away which contains the number with the smallest digit sum. If two or more neighbors exist with the same digit sum then from those squares the one with the lowest number is chosen.
%C A343563 The sequence is finite. After 790 steps the square with number 69 is visited, after which all eight neighboring squares have been visited. The largest visit spiral number is a(626) = 6112, while there are four squares with the largest visited digit sum of 19: a(373) = 2683, a(539) = 2737, a(590) = 2944, a(594) = 2728.
%H A343563 Scott R. Shannon, <a href="/A343563/a343563.png">Image showing the 791 visited squares</a>. The starting square is highlighted in white, the visited squares in yellow, the final square in red, while the path is colored across the spectrum to show the relative step ordering.
%H A343563 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).
%e A343563 The board is numbered with the square spiral:
%e A343563 .
%e A343563   17--16--15--14--13   .
%e A343563    |               |   .
%e A343563   18   5---4---3  12   29
%e A343563    |   |       |   |   |
%e A343563   19   6   1---2  11   28
%e A343563    |   |           |   |
%e A343563   20   7---8---9--10   27
%e A343563    |                   |
%e A343563   21--22--23--24--25--26
%e A343563 .
%e A343563 a(2) = 10 as the eight unvisited neighbors of the square a(1) = 1 are numbered 10,12,14,16,18,20,22,24, and 10, with a digit sum of 1, has the lowest digit sum of these.
%e A343563 a(4) = 30 as the seven unvisited neighbors of the square a(3) = 3 square are numbered 6,8,28,30,32,34,16, and 30, with a digit sum of 3, has the lowest digit sum of these.
%e A343563 a(9) = 5 as two of the unvisited neighbors of the square a(8) = 2 are 5 and 23, both of which have a digit sum of 5, but 5 is chosen as it is the lower number.
%Y A343563 Cf. A007953, A316667, A328894, A326922, A328928.
%K A343563 nonn,base,fini
%O A343563 1,2
%A A343563 _Scott R. Shannon_, Apr 19 2021