A316667 Squares visited by a knight moving on a spirally numbered board always to the lowest available unvisited square.

1, 10, 3, 6, 9, 4, 7, 2, 5, 8, 11, 14, 29, 32, 15, 12, 27, 24, 45, 20, 23, 44, 41, 18, 35, 38, 19, 16, 33, 30, 53, 26, 47, 22, 43, 70, 21, 40, 17, 34, 13, 28, 25, 46, 75, 42, 69, 104, 37, 62, 95, 58, 55, 86, 51, 48, 77, 114, 73, 108, 151, 68, 103, 64, 67, 36

Offset

1,2

Comments

Board is numbered with the square spiral:

.

17--16--15--14--13 .

| | .

18 5---4---3 12 .

| | | | .

19 6 1---2 11 .

| | | .

20 7---8---9--10 .

| .

21--22--23--24--25--26

.

This sequence is finite: At step 2016, square 2084 is visited, after which there are no unvisited squares within one knight move.

Links

Daniël Karssen, Table of n, a(n) for n = 1..2016

Daniël Karssen, Figure showing the first 60 steps of the sequence

Daniël Karssen, Figure showing the complete sequence

Daniël Karssen, MATLAB script to generate the complete sequence

N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019)

Formula

a(n) = A316328(n-1) + 1.

Prog

(PARI) A316667(n)=A316328(n-1)+1 \\ M. F. Hasler, Nov 06 2019

Crossrefs

Cf. A316328 (same starting at 0), A329022 (same with diamond-shaped spiral), A316588 (variant on board with x,y >= 0).

Cf. A326924 (choose square closest to the origin), A328908 (using taxicab distance), A328909 (using sup norm); A323808, A323809.

The (x,y) coordinates of square k are (A174344(k), A274923(k)).

Sequence in context: A241887 A182493 A323763 * A329518 A329519 A323808

Adjacent sequences: A316664 A316665 A316666 * A316668 A316669 A316670

Keyword

nonn,fini,full,look

Author

Daniël Karssen, Jul 10 2018, following a suggestion from N. J. A. Sloane, Jul 09 2018

Status

approved