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%I #46 Mar 07 2020 13:50:20
%S 0,1,0,1,2,3,0,2,3,1,2,3,0,2,3,1,4,5,6,7,0,4,5,6,7,1,4,5,6,7,0,4,5,6,
%T 7,1,2,5,4,7,6,3,0,2,5,4,7,6,3,1,2,5,4,7,6,3,0,2,5,4,7,6,3,1,8,9,10,
%U 11,12,13,14,15,0,8,9,10,11,12,13,14,15,1,8
%N Walk a rook along the square spiral numbered 0, 1, 2, ... (cf. A274641); a(n) = mex of earlier values the rook can move to.
%C Analog of A308884 but using a rook rather than a knight.
%C The array of values - see the illustration in the link - appears to have a number of interesting symmetries.
%H Rémy Sigrist, <a href="/A308896/b308896.txt">Table of n, a(n) for n = 0..16128</a>
%H F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i1p52/8039">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H Rémy Sigrist, <a href="/A308896/a308896_1.png">Colored representation of the spiral for -511 <= x, y <= 511</a> (where dark pixels correspond to higher values and red pixels correspond to 0's)
%H Rémy Sigrist, <a href="/A308896/a308896_3.png">Scatterplot of (x,y) such that A(x,y) has bit b set to one for b = 0..6 and -63 <= x <= 64 and -63 <= y <= 64</a>
%H Rémy Sigrist, <a href="/A308896/a308896.gp.txt">PARI program for A308896</a>
%H N. J. A. Sloane, <a href="/A308896/a308896.png">Initial terms of spiral</a>
%H N. J. A. Sloane, <a href="/A308896/a308896_2.txt">Explicit formulas for the array in A308896</a>, Jul 02 2019
%H N. J. A. Sloane, <a href="/A308896/a308896_2.png">The two kinds of sectors.</a> (Rows y=1 and above form a sector of the first type, rows y=0 and below form the second type.)
%F a(n) = 0, 1 iff n belongs to A002378, A085046, respectively. - _Rémy Sigrist_, Jul 02 2019
%F For formulas for the terms in the array, see the "Explicit formulas" link.
%e The central 21 X 21 portion of the plane:
%e [ 4 1 3 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17 2 0]
%e [ 5 2 1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 0 3]
%e [18 17 16 1 3 6 7 12 13 14 15 8 9 10 11 4 5 2 0 31 30]
%e [19 16 17 2 1 7 6 13 12 15 14 9 8 11 10 5 4 0 3 30 31]
%e [16 19 18 5 4 1 3 14 15 12 13 10 11 8 9 2 0 7 6 29 28]
%e [17 18 19 4 5 2 1 15 14 13 12 11 10 9 8 0 3 6 7 28 29]
%e [22 21 20 11 10 9 8 1 3 6 7 4 5 2 0 15 14 13 12 27 26]
%e [23 20 21 10 11 8 9 2 1 7 6 5 4 0 3 14 15 12 13 26 27]
%e [20 23 22 9 8 11 10 5 4 1 3 2 0 7 6 13 12 15 14 25 24]
%e [21 22 23 8 9 10 11 4 5 2 1 0 3 6 7 12 13 14 15 24 25]
%e *26 25 24 15 14 13 12 7 6 3 *0* 1 2 5 4 11 10 9 8 23 22]
%e [27 24 25 14 15 12 13 6 7 0 2 3 1 4 5 10 11 8 9 22 23]
%e [24 27 26 13 12 15 14 3 0 4 5 6 7 1 2 9 8 11 10 21 20]
%e [25 26 27 12 13 14 15 0 2 5 4 7 6 3 1 8 9 10 11 20 21]
%e [30 29 28 7 6 3 0 8 9 10 11 12 13 14 15 1 2 5 4 19 18]
%e [31 28 29 6 7 0 2 9 8 11 10 13 12 15 14 3 1 4 5 18 19]
%e [28 31 30 3 0 4 5 10 11 8 9 14 15 12 13 6 7 1 2 17 16]
%e [29 30 31 0 2 5 4 11 10 9 8 15 14 13 12 7 6 3 1 16 17]
%e [ 6 3 0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 2]
%e [ 7 0 2 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30 3 1]
%e [ 0 4 5 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29 6 7]
%e ===============================**===============================
%o (PARI) See Links section.
%Y Cf. A002378, A003987, A085046, A274641, A308884, A308897.
%K nonn,look
%O 0,5
%A _N. J. A. Sloane_, Jul 02 2019
%E More terms from _Rémy Sigrist_, Jul 02 2019
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