A395506 Two players, Black and Red, take turns on a square spiral. 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. We then consider the cells that are eventually occupied: a(n) tells at which turn the n-th such cell has been occupied.

1, 2, 3, 4, 6, 5, 8, 7, 10, 9, 12, 11, 13, 15, 14, 16, 17, 19, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 28, 31, 30, 33, 32, 34, 35, 36, 38, 37, 40, 39, 42, 41, 43, 45, 44, 47, 49, 46, 51, 48, 53, 50, 55, 52, 57, 54, 59, 56, 61, 58, 60, 63, 65, 67, 69, 71, 62

Offset

1,2

Comments

A permutation of the positive integers with inverse A395507.

The sequence can be built from the nonzero values of A395486: replace each positive value v by 2*v-1, and each negative value v by -2*v.

Links

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

Rémy Sigrist, PARI program

Index entries for sequences that are permutations of the natural numbers

Example

The first terms alongside the square spiral are:
     .--63---.---.--60--58---.--61---.---.--56
     |                                       |
     .  38--36---.--35---.---.--34--32--33  59
     |   |                               |   |
     .  37  21--20--18---.---.--19--17  30  54
     |   |   |                       |   |   |
     .  40  22   .--11---.---.--12   .   .  57
     |   |   |   |               |   |   |   |
     .  39   .   .   6---4---3   9   .   .  52
     |   |   |   |   |       |   |   |   |   |
     .  42   .   .   5   1---2  10   .   .  55
     |   |   |   |   |           |   |   |   |
     .  41  23   .   8---.---.---7   .   .  50
     |   |   |   |                   |   |   |
     .  43  24  13--15---.---.--14--16  31  53
     |   |   |                           |   |
     .  45  25---.--26---.---.--27--29--28  48
     |   |                                   |
    65  44---.---.--47--49---.--46---.---.--51
     |
    67--69--71---.--62--64--73--75---.---.--66

Prog

(PARI) \\ See Links section.
(C#) // Use program provided in A392177 with the 3 arguments "- - occupation".

Crossrefs

Cf. A392177, A392178, A395355, A395486, A395507 (inverse).

Sequence in context: A123717 A123718 A335430 * A395507 A121878 A167905

Adjacent sequences: A395503 A395504 A395505 * A395507 A395508 A395509

Keyword

nonn

Author

Rémy Sigrist, Apr 26 2026

Status

approved