A393236 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 the number of pieces among the first n with a record index when placed, minus the others.

1, 2, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 15, 16, 15, 16, 15, 16, 17, 18, 19, 18, 19, 18, 19, 18, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20, 19, 20

Offset

1,2

Links

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

Rémy Sigrist, Scatterplot of the first 1000000 terms

Rémy Sigrist, PARI program

Formula

a(n+2) >= a(n).

Example

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

Prog

(PARI) \\ See Links section.

Crossrefs

Cf. A392177, A395355, A395818.

Sequence in context: A017850 A305901 A291520 * A354445 A317646 A305748

Adjacent sequences: A393233 A393234 A393235 * A393237 A393238 A393239

Keyword

nonn

Author

Rémy Sigrist, May 07 2026

Status

approved