A392183 - OEIS

A392183

Consider the square spiral in the first quadrant, with its cells numbered starting at 0, as in A392181. Two players, Black and Red, take turns. When it is Black's turn, he places a knight at the smallest unoccupied cell not attacked by an existing Red knight, and when it is Red's turn, she places a knight at the smallest unoccupied cell not attacked by an existing Black knight. Sequence lists squares occupied by a Red knight.

%I #20 Feb 15 2026 10:15:26

%S 1,3,4,14,19,20,21,24,30,35,36,39,41,43,45,47,55,56,57,65,68,72,81,83,

%T 85,87,89,91,93,100,105,109,110,111,127,130,132,136,145,149,151,153,

%U 155,157,158,159,161,167,171,175,181,182,183,184,186,187,189,201,203,204

%N Consider the square spiral in the first quadrant, with its cells numbered starting at 0, as in A392181. Two players, Black and Red, take turns. When it is Black's turn, he places a knight at the smallest unoccupied cell not attacked by an existing Red knight, and when it is Red's turn, she places a knight at the smallest unoccupied cell not attacked by an existing Black knight. Sequence lists squares occupied by a Red knight.

%C There is a subtle point about the definition. Because the players take turns in placing a knight, the spiral is not filled in consecutively. To obtain the sequence, one must first fill in the whole spiral, and then return to the start, and record the Red knights that are encountered as one walks along the spiral.

%H Michael S. Branicky, <a href="/A392183/b392183.txt">Table of n, a(n) for n = 1..50000</a>

%H Michael S. Branicky, <a href="/A392182/a392182.py.txt">Python program for OEIS A392182 and A392183</a>

%H Michael S. Branicky, <a href="/A392182/a392182.png">The first 10^4 filled cells of the spiral</a>

%H Michael S. Branicky, <a href="/A392182/a392182_1.png">The first 10^6 filled cells of the spiral</a>

%H Michael S. Branicky, <a href="/A392182/a392182_2.png">The first 4*10^6 filled cells of the spiral</a>

%H Michael S. Branicky, <a href="/A392182/a392182_3.png">The first 64*10^6 filled cells of the spiral</a>

%H N. J. A. Sloane, <a href="/A392181/a392181.jpg">The playing field: The square spiral in the first quadrant, with initial square labeled 0.</a>

%H N. J. A. Sloane, <a href="/A392182/a392182.jpg">Squares 0 to 100 of the spiral.</a> [Black circle = Black knight, red circle = Red knight, black slash = attacked by a Black knight, red slash = attacked by a Red knight.]

%o (Python) # see linked program

%Y Cf. A392177-A392182.

%K nonn

%O 1,2

%A _Michael S. Branicky_ and _N. J. A. Sloane_, Feb 14 2026