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A383185 Number of the square visited by a king moving on a spirally numbered board always to the lowest available unvisited square, when a wall delimiting the spiral must be crossed on each move.

%I A383185 #38 May 21 2025 11:21:21
%S A383185 0,3,13,2,10,1,7,21,6,18,4,14,32,12,28,11,27,9,23,8,22,44,20,40,19,5,
%T A383185 17,37,16,34,15,33,59,31,57,30,54,29,53,85,51,25,47,24,46,76,45,75,43,
%U A383185 73,42,70,41,69,39,67,38,66,36,62,35,61,95,60,94,58,92,56,88,55,87,127,86,52,26
%N A383185 Number of the square visited by a king moving on a spirally numbered board always to the lowest available unvisited square, when a wall delimiting the spiral must be crossed on each move.
%C A383185 The board is numbered following a square spiral starting with 0 at the origin (where the king is at n = 0) and delimited by a wall that must be crossed on each move:
%C A383185 .
%C A383185   16  15  14  13  12 | .
%C A383185      ,-----------.   | .
%C A383185   17 | 4   3   2 |11 | .
%C A383185      |   ,---    |   | .
%C A383185   18 | 5 | 0   1 |10 | .
%C A383185      |   '-------'   | .
%C A383185   19 | 6   7   8   9 | .
%C A383185      `---------------' .
%C A383185   20  21  22  23  24  25
%C A383185 .
%C A383185 A line drawn from the center of the starting square to the center of the ending square must pass through a wall on each move. A move that would just touch a wall without passing through the wall (e.g., 0 to 2) is not permissible. Equivalently, the king can't move from a square labeled k to a square labeled k +- 1 or k +- 2, i.e., |a(n)-a(n+1)| > 2 for all n.
%C A383185 This sequence is a permutation of the nonnegative integers, see A383186 for the inverse permutation. The king's walk indeed fills the 2D grid with an initial segment S0 of 24 moves, followed by rings R(r), r >= 1, which consist of three shells S1(r), S2(r) and S3(r), each of which corresponds to a tour around the center. Each ring R(r) starts with the move number n = 48 r^2 - 16 r + 1 = (25, 147, 365, ...) to the square at position P(r) = (2-3r, 3r-3) = ((-1,0), (-4,3), (-7,6), ...), and contains a perfectly well defined sequence of 96 r + 26 grid points following a precise sequence of pattern given in full detail on the wiki page provided in the links section.
%H A383185 M. F. Hasler, <a href="/A383185/b383185.txt">Table of n, a(n) for n = 0..1000</a>
%H A383185 M. F. Hasler, <a href="/wiki/User:M._F._Hasler/A383185">A383185: Details about the grid filling king's walk</a>, OEIS wiki, May 15, 2025.
%H A383185 M. F. Hasler, <a href="/A383185/a383185.pdf">Path plot with annotations</a>, May 15, 2025.
%H A383185 <a href="/index/Per#IntegerPermutation">OEIS index to sequences related to permutations of the integers</a>.
%e A383185 For n = 1, a(1) = 3 because moving from 0 to 1 or 2 does not pass through a wall.
%o A383185 (Python)
%o A383185 def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real)
%o A383185     else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y)
%o A383185 def A383185(n):
%o A383185     if not hasattr(A:=A383185, 'terms'): A.terms=[0]; A.pos=0; A.path=[0]
%o A383185     while len(A.terms) <= n:
%o A383185         s,d = min((s,d) for d in (1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j) if
%o A383185             abs((s:=square_number(A.pos+d))-A.terms[-1]) > 2 and s not in A.terms)
%o A383185         A.terms.append(s); A.pos += d; A.path.append(A.pos)
%o A383185     return A.terms[n]
%o A383185 import matplotlib.pyplot as plt # this and below to plot the trajectory
%o A383185 plt.plot([z.real for z in A383185.path], [z.imag for z in A383185.path])
%o A383185 plt.show()
%Y A383185 Cf. A375925 (the same with indices and numbers of squares starting at 1).
%Y A383185 Cf. A383186 (inverse permutation).
%Y A383185 Cf. A316328 (knight's path), A033638, A316667 (trapped knight), A336038 (trapped king), A335856 (trapped king preferably moving to prime numbers).
%K A383185 nonn,walk
%O A383185 0,2
%A A383185 _M. F. Hasler_, May 12 2025