A323749 - OEIS

A323749

Triangle read by rows: T(n,m) (1 <= n < m) = number of moves of a (m,n)-leaper (a generalization of a chess knight) until it can no longer move, starting on a board with squares spirally numbered from 1. Each move is to the lowest-numbered unvisited square. T(n,m) = -1 if the path never terminates.

%I #23 Dec 10 2023 17:56:00

%S 2016,3723,4634,13103,2016,1888,14570,7574,1323,4286,26967,3723,2016,

%T 4634,1796,101250,12217,4683,9386,1811,3487,158735,13103,5974,2016,

%U 2758,1888,3984,132688,33864,3723,8900,6513,4634,4505,7796,220439,14570,36232,7574,2016,1323,9052,4286,5679,144841,52738,19370,6355,6425

%N Triangle read by rows: T(n,m) (1 <= n < m) = number of moves of a (m,n)-leaper (a generalization of a chess knight) until it can no longer move, starting on a board with squares spirally numbered from 1. Each move is to the lowest-numbered unvisited square. T(n,m) = -1 if the path never terminates.

%C The entries are the lower triangle of an array, for an (m,n)-leaper, where 1 <= n < m, ordered: (2,1), (3,1), (3,2), (4,1), (4,2), etc. Are all the paths finite? This appears to be an open question.

%H Jud McCranie, <a href="/A323749/b323749.txt">Table of n, a(n) for n = 1..19900</a>

%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 A chess knight (a (2,1)-leaper) makes 2016 moves before it has no moves available (see A316667). Initial placement on square 1 counts as one move.

%Y Cf. A316667, A323750, A317106, A317471, A317416, A323750, A317438, A317916.

%K nonn,tabf

%O 1,1

%A _Jud McCranie_, Jan 26 2019

%E Edited by _N. J. A. Sloane_, Apr 30 2021