A396185 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 first cell in the spiral that is beyond the last Red knight and is not attacked by any Red knight, and when it is Red's turn, she places a knight at the first cell that is beyond the last Black knight and is not not attacked by any Black knight. Sequence lists squares occupied by Black knights.

0, 2, 5, 7, 13, 29, 36, 40, 42, 47, 49, 54, 56, 61, 63, 67, 69, 71, 75, 79, 85, 91, 93, 100, 115, 120, 122, 127, 142, 144, 148, 150, 152, 154, 156, 160, 165, 169, 173, 177, 180, 182, 189, 191, 193, 195, 199, 203, 205, 207, 209, 213, 216, 231, 235, 239, 243, 245, 250, 272, 281, 286, 288, 292, 294, 297, 306, 314, 318, 322, 324, 328

Offset

0,2

Comments

Similar to A392177, except that now the Black and Red knights must alternate along the spiral. In A392177 it was possible to have two successive knights of the same color, but here that is forbidden.

References

Ingmar Eissfeldt, Email to N. J. A. Sloane, May 15 2026.

Links

Ingmar Eissfeldt, Table of n, a(n) for n = 0..19999

Michael S. Branicky, Python program for OEIS A396185, A396186 and A396187

Ingmar Eissfeldt, Comments on the Spiral

Ingmar Eissfeldt, Full gallery

Ingmar Eissfeldt, The first 10^4 cells of the spiral. To locate the 0 cell of the spiral, after magnification there is a unique 2X2 block of cells which are all black except for the bottom left cell which is red. The three black cells are cells 5, 0, and 7, and the red cell is cell 6.

Ingmar Eissfeldt, The first 10^5 cells of the spiral

Ingmar Eissfeldt, The first 10^6 cells of the spiral

Ingmar Eissfeldt, The first 10^7 cells of the spiral [This file may not open in your browser. To view it, save it and open it on your computer.]

Ingmar Eissfeldt, The first 10^8 cells of the spiral [This file may not open in your browser. To view it, save it and open it on your computer.]

Ingmar Eissfeldt, The first 10^9 cells of the spiral [This file may not open in your browser. To view it, save it and open it on your computer.]

Ingmar Eissfeldt, Gallery of features

Thorsten H., Gallery of variant with 3 knights instead of 2

N. J. A. Sloane, Sketch of cells 0-121. (Black circles = Black knights, red circles = Red knights, black (resp. red) slashes = attacked by a Black (resp. Red) knight.)

Crossrefs

Cf. A274641, A308884, A392177, A392178, A396186, A396187.

Sequence in context: A212319 A161379 A250173 * A038945 A215210 A177997

Adjacent sequences: A396182 A396183 A396184 * A396186 A396187 A396188

Keyword

nonn

Author

N. J. A. Sloane, May 31 2026, based on an email from Ingmar Eißfeldt, May 15 2026

Status

approved