A392178 - OEIS

A392178

Consider the square spiral with its cells numbered starting at 0, as in A308884 and A274641. 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.

1, 3, 4, 6, 10, 12, 24, 25, 34, 35, 37, 41, 44, 49, 55, 57, 58, 63, 64, 66, 68, 72, 78, 82, 84, 86, 88, 90, 95, 96, 114, 115, 120, 121, 127, 136, 137, 139, 141, 142, 143, 145, 147, 149, 155, 164, 167, 169, 171, 175, 181, 184, 189, 190, 195, 196, 198, 200, 202, 204, 210, 216, 221, 222, 223, 224, 225, 226, 227, 228, 230, 232, 234, 236

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Please see the Comments in A392177, which will not be repeated here.

REFERENCES

Michael S. Branicky, Jonas Karlsson, and N. J. A. Sloane, Proof of convergence of the Red and Black Knights structure, MS in preparation, May 14 2026.

Jonas Karlsson, Letter to N. J. A. Sloane, Jan 23 2026, and emails from Feb 05-06 2026.

LINKS

Jonas Karlsson, Table of n, a(n) for n = 1..45763 [More than the usual number of terms are shown, in view of the complexity of the sequence.]

Michael S. Branicky, The first 10^6 cells of the spiral.

Michael S. Branicky, The first 4*10^6 cells of the spiral

Michael S. Branicky, The first 16*10^6 cells of the spiral

Michael S. Branicky, The first 64*10^6 cells of the spiral

Michael S. Branicky, The first 256*10^6 cells of the spiral

Michael S. Branicky, Python program for OEIS A392177 and A392178

Brady Haran, Jonas Karlsson, and N. J. A. Sloane, Red & Black Knights (extraordinary result); Amazing Chessboard Patterns (extra), YouTube Numberphile videos, May 2026

Jonas Karlsson, The first 1000 cells of the spiral.

Jonas Karlsson, The first 100000 cells of the spiral.

Jonas Karlsson, Table of a(n) for n = 1..10^7 [95 MB file]

N. J. A. Sloane, The first five layers of the spiral. [Black circle = Black knight, red circle = Red knight, black slash = attacked by a Black knight, red slash = attacked by a Red knight.] Note that the links are deliberately arranged in logical order, not alphabetical order (see Comments in A392177)

PROG

(Python) # see linked program

CROSSREFS

Cf. A274641, A308884, A308885 (if all the knights are of the same color), A392177, A392179, A392180.

Sequence in context: A113887 A103000 A344516 * A383387 A310001 A310002

Adjacent sequences: A392175 A392176 A392177 * A392179 A392180 A392181

KEYWORD

nonn,look,changed

AUTHOR

N. J. A. Sloane, Feb 06 2026, based on a letter and emails from Jonas Karlsson.

STATUS

approved