A326413 - OEIS

%I #90 Feb 01 2022 00:31:36

%S 0,0,1,0,1,0,1,1,2,2,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,2,1,1,0,1,1,1,1,

%T 1,1,0,1,1,1,0,2,2,1,1,0,2,3,2,2,1,3,1,1,1,1,1,2,2,3,2,3,1,4,3,5,6,1,

%U 1,1,1,1,3,1,1,1,3,1,1,0,1,1,1,1,1,1,1,1,2,0,0,1,1,0,0,1,0,1,1,1,0,1,0

%N Successive squares visited by a knight on the single-digit square spiral, with ties resolved towards the left.

%C Take the standard counterclockwise square spiral starting at 0, as in A304586, but only write one digit at a time in the cells of the spiral: 0,1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,...

%C Place a chess knight at cell 0. Move it to the lowest-numbered cell it can attack, and if there is a tie, move it to the cell closest (in Euclidean distance) to the start, and if there is still a tie, move to the left(*).

%C No cell can be visited more than once.

%C Inspired by the Trapped Knight video and A316667.

%C Just as for A316667, the sequence is finite. After a while, the knight has no unvisited squares it can reach, and the sequence ends with a(1217) = 4.

%C (*)Moving to the left means choose the point with the lowest x-coordinate. This leads to an unambiguous choice of tied squares only for the 'move left' case.

%H Luca Petrone, <a href="/A326413/b326413.txt">Table of n, a(n) for n = 1..1217</a>

%H Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2019/05/kneils-knumberphile-knight.html">Kneil's Knumberphile Knight</a>, Cinquante signes, May 04 2019.

%H M. F. Hasler, <a href="/wiki/Knight_tours">Knight tours</a>, OEIS wiki, Nov. 2019.

%H N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=RGQe8waGJ4w">The Trapped Knight</a>, Numberphile video (2019).

%e The digit-square spiral is

%e                                 .

%e                                 .

%e     2---2---2---1---2---0---2   2

%e     |                       |   |

%e     3   1---2---1---1---1   9   3

%e     |   |               |   |   |

%e     2   3   4---3---2   0   1   1

%e     |   |   |       |   |   |   |

%e     4   1   5   0---1   1   8   3

%e     |   |   |           |   |   |

%e     2   4   6---7---8---9   1   0

%e     |   |                   |   |

%e     5   1---5---1---6---1---7   3

%e     |                           |

%e     2---6---2---7---2---8---2---9

%Y Cf. A304586, A316667, A328698.

%Y Cf. A326916, A326918; A316328; A326924, A326922; A328908, A328928; A328909, A328929.

%K nonn,fini,full

%O 1,9

%A _N. J. A. Sloane_, Oct 17 2019

%E More terms from _Luca Petrone_

%E Corrected and extended by _Eric Angelini_, Oct 24 2019