cp's OEIS Frontend

A352731 On a diagonally numbered square grid, with labels starting at 1, this is the number of the last cell that a (1,n) leaper reaches before getting trapped when moving to the lowest available unvisited square, or -1 if it never gets trapped.

%I A352731 #15 Mar 29 2023 09:20:24
%S A352731 -1,1378,-1,595,66,36,153,758,1185,78,1732,171,2510,2094,1407,253,630,
%T A352731 210,780,2385,1326,300,1225,990,2800,406,3267,4333,4124,528,4309,741,
%U A352731 5951,666,2701,903,30418,820,3321,1081,4186,990,8299,2775,4560,1176,4753,39951,5778
%N A352731 On a diagonally numbered square grid, with labels starting at 1, this is the number of the last cell that a (1,n) leaper reaches before getting trapped when moving to the lowest available unvisited square, or -1 if it never gets trapped.
%C A352731 A (1,2) leaper is a chess knight. (1,1) and (1,3) leapers both never get trapped. This is understandable for the (1,1) leaper but not so much for the (1,3) which does get trapped on the spirally numbered board (see A323471). Once the (1,3) leaper reaches 39 it then performs the same set of 4 moves repeatedly, meaning that it never gets trapped.
%o A352731 (Python)
%o A352731 # reformatted by _R. J. Mathar_, 2023-03-29
%o A352731 class A352731():
%o A352731     def __init__(self,n) :
%o A352731         self.n = n
%o A352731         self.KM=[(n, 1), (1, n), (-1, n), (-n, 1), (-n, -1), (-1, -n), (1, -n), (n, -1)]
%o A352731     @staticmethod
%o A352731     def _idx(loc):
%o A352731         i, j = loc
%o A352731         return (i+j-1)*(i+j-2)//2 + j
%o A352731     def _next_move(self,loc, visited):
%o A352731         i, j = loc
%o A352731         moves = [(i+io, j+jo) for io, jo in self.KM if i+io>0 and j+jo>0]
%o A352731         available = [m for m in moves if m not in visited]
%o A352731         return min(available, default=None, key=lambda x: A352731._idx(x))
%o A352731     def _aseq(self):
%o A352731         locs = [[], []]
%o A352731         loc, s, turn, alst = [(1, 1), (1, 1)], {(1, 1)}, 0, [1]
%o A352731         m = self._next_move(loc[turn], s)
%o A352731         while m != None:
%o A352731             loc[turn], s, turn, alst = m, s|{m}, 0 , alst + [A352731._idx(m)]
%o A352731             locs[turn] += [loc[turn]]
%o A352731             m = self._next_move(loc[turn], s)
%o A352731             if len(s)%100000 == 0:
%o A352731                 print(self.n,'{steps} moves in'.format(steps = len(s)))
%o A352731         return alst
%o A352731     def at(self,n) :
%o A352731         if n == 1 or n == 3:
%o A352731             return -1
%o A352731         else:
%o A352731             return self._aseq()[-1]
%o A352731 for n in range(1,40):
%o A352731     a352731 = A352731(n)
%o A352731     print(a352731.at(n))
%Y A352731 Cf. A323469, A323471, A352730.
%K A352731 sign
%O A352731 1,2
%A A352731 _Andrew Smith_, Mar 30 2022