A308896 Walk a rook along the square spiral numbered 0, 1, 2, ... (cf. A274641); a(n) = mex of earlier values the rook can move to.

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, 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, 11, 12, 13, 14, 15, 0, 8, 9, 10, 11, 12, 13, 14, 15, 1, 8

Offset: 0

Author

N. J. A. Sloane, Jul 02 2019

Keywords

• nonn
• look

Comments

Analog of A308884 but using a rook rather than a knight.

The array of values - see the illustration in the link - appears to have a number of interesting symmetries.

Examples

			The central 21 X 21 portion of the plane:
[ 4  1  3 30 31 28 29 26 27 24 25 22 23 20 21 18 19 16 17  2  0]
[ 5  2  1 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16  0  3]
[18 17 16  1  3  6  7 12 13 14 15  8  9 10 11  4  5  2  0 31 30]
[19 16 17  2  1  7  6 13 12 15 14  9  8 11 10  5  4  0  3 30 31]
[16 19 18  5  4  1  3 14 15 12 13 10 11  8  9  2  0  7  6 29 28]
[17 18 19  4  5  2  1 15 14 13 12 11 10  9  8  0  3  6  7 28 29]
[22 21 20 11 10  9  8  1  3  6  7  4  5  2  0 15 14 13 12 27 26]
[23 20 21 10 11  8  9  2  1  7  6  5  4  0  3 14 15 12 13 26 27]
[20 23 22  9  8 11 10  5  4  1  3  2  0  7  6 13 12 15 14 25 24]
[21 22 23  8  9 10 11  4  5  2  1  0  3  6  7 12 13 14 15 24 25]
*26 25 24 15 14 13 12  7  6  3 *0* 1  2  5  4 11 10  9  8 23 22]
[27 24 25 14 15 12 13  6  7  0  2  3  1  4  5 10 11  8  9 22 23]
[24 27 26 13 12 15 14  3  0  4  5  6  7  1  2  9  8 11 10 21 20]
[25 26 27 12 13 14 15  0  2  5  4  7  6  3  1  8  9 10 11 20 21]
[30 29 28  7  6  3  0  8  9 10 11 12 13 14 15  1  2  5  4 19 18]
[31 28 29  6  7  0  2  9  8 11 10 13 12 15 14  3  1  4  5 18 19]
[28 31 30  3  0  4  5 10 11  8  9 14 15 12 13  6  7  1  2 17 16]
[29 30 31  0  2  5  4 11 10  9  8 15 14 13 12  7  6  3  1 16 17]
[ 6  3  0 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  1  2]
[ 7  0  2 17 16 19 18 21 20 23 22 25 24 27 26 29 28 31 30  3  1]
[ 0  4  5 18 19 16 17 22 23 20 21 26 27 24 25 30 31 28 29  6  7]
===============================**===============================
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..16128
• F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
• Rémy Sigrist, Colored representation of the spiral for -511 <= x, y <= 511 (where dark pixels correspond to higher values and red pixels correspond to 0's)
• Rémy Sigrist, 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
• Rémy Sigrist, PARI program for A308896
• N. J. A. Sloane, Initial terms of spiral
• N. J. A. Sloane, Explicit formulas for the array in A308896, Jul 02 2019
• N. J. A. Sloane, The two kinds of sectors. (Rows y=1 and above form a sector of the first type, rows y=0 and below form the second type.)

Crossrefs

Cf. A002378, A003987, A085046, A274641, A308884, A308897.

Programs

PARI

See Links section.

Formula

a(n) = 0, 1 iff n belongs to A002378, A085046, respectively. - Rémy Sigrist, Jul 02 2019

For formulas for the terms in the array, see the "Explicit formulas" link.

Extensions

More terms from Rémy Sigrist, Jul 02 2019