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

Author

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

Keywords

• nonn
• fini
• full
• look

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)

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)).

Programs

PARI

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

Formula

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