A395818 Consider the square spiral with its cells numbered starting at 0. 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. a(n) gives, among the first n pieces, the number of Black pieces with a record index when placed minus the number of Red pieces with a record index when placed.

1, 0, 1, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 4, 3, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 1, 1, 0, 0, -1, 0, -1, 0, 0, 1, 1, 2, 2, 3, 2, 2, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, -7, -7, -8, -8, -9, -9, -10, -10, -11, -11, -12, -12, -13, -13

Offset

1,7

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, Scatterplot of the first 1000000 terms

Rémy Sigrist, PARI program

Example

The first terms are:
  n   a(n)  |A395355(n-1)|  Color  Record?
  --  ----  --------------  -----  -------
   1     1               0  Black  Yes
   2     0               1  Red    Yes
   3     1               2  Black  Yes
   4     0               3  Red    Yes
   5     1               5  Black  Yes
   6     1               4  Red    No
   7     2               9  Black  Yes
   8     2               6  Red    No
   9     3              11  Black  Yes
  10     3              10  Red    No
  11     4              15  Black  Yes
  12     4              12  Red    No
  13     5              20  Black  Yes
  14     4              24  Red    Yes
  15     4              21  Black  No

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A392177, A392180, A393449, A393236, A395355.

Sequence in context: A187446 A240020 A336430 * A368086 A167232 A319468

Adjacent sequences: A395814 A395816 A395817 * A395819 A395820 A395823

Keyword

sign

Author

Rémy Sigrist, May 07 2026

Status

approved