A316667 -id:A316667 - cp's OEIS frontend

A336038 Squares visited by a chess king on a square-spiral numbered board and stepping to the lowest unvisited adjacent square, where each step is not in the same direction as the previous step.

1, 2, 3, 4, 6, 5, 15, 14, 12, 11, 9, 8, 22, 7, 19, 18, 16, 17, 35, 34, 60, 32, 13, 29, 28, 10, 25, 24, 46, 23, 45, 21, 20, 40, 39, 67, 37, 36, 38, 66, 64, 63, 97, 61, 62, 96, 95, 59, 33

Offset: 1

Scott R. Shannon, Jul 12 2020

This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step, which is not in the same direction as its previous step, moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the lowest spiral number.
The sequence is finite. After 48 steps the square with spiral number 33 is reached after which all eight adjacent squares have been visited.
If the king simply moved to the lowest numbered unvisited adjacent square the walk would be infinite as the king would just follow the path of the square spiral. By not allowing consecutive moves in the same direction forces the king off this minimal numbered path. The first time this happens is a(5) = 6 as from a(4) = 4 the lowest numbered adjacent square is 5 but that would require a step directly to the left, the same as the previous step from a(3) = 3 to a(4).

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square of the king.
a(2) = 2. The eight adjacent unvisited squares around a(1) are numbered 2,3,4,5,6,7,8,9. Of these 2 is the lowest.
a(5) = 6. The five adjacent unvisited squares around a(4) = 4 are numbered 5,6,14,15,16. Of these 5 is the lowest but that would require a step directly left from 4, which is the same step as a(3) = 3 to a(4) = 4, so is not allowed. The next lowest available square is 6.

Scott R. Shannon, Image showing the 48 steps of the king's path. The green dot is the first square with number 1 and the red dot the last square with number 33. The red dot is surrounded by blue dots to show the eight occupied squares. The yellow dots marks the smallest unvisited square with number 26.
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A336208, A335816, A333713, A333714, A316667.

A343530 Number of steps before being trapped for a knight moving on a square-spiral base-n numbered board when stepping to the closest unvisited square which contains a number that shares no digit with the number of the current square. If two or more such squares are the same distance away the one with the smaller number is chosen.

Original entry on oeis.org

0, 1, 12, 10, 13, 16, 35, 51, 56, 90, 42, 84, 99, 129, 156, 30, 220, 184, 201, 79, 321, 25, 424, 301, 389, 29, 32, 311, 328, 186, 129, 42, 101, 97, 144, 52, 534, 83, 506, 885, 233, 472, 43, 410, 145, 210, 482, 51, 57, 144, 53, 60, 148, 248, 83, 80, 180, 72, 55

Offset: 2

Scott R. Shannon and Eric Angelini, Apr 19 2021

			The board in base 10 is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12  29
   |   |       |   |   |
  19   6   1---2  11  28
   |   |           |   |
  20   7---8---9--10  27
   |                   |
  21--22--23--24--25--26
.
a(2) = 0 as on a base-2 numbered spiral all surrounding squares contain a 1 digit in their number thus, as the knight starts on the square numbered 1, it has no square to move to which does not contain a 1 digit.
a(3) = 1 as on a base-3 numbered board there are two squares the knight can step to which do not contain a 1 digit -- the squares numbered 200_3 = 18 and 220_3 = 24. The knight steps to 200_3 as it is the lowest numbered square, but after that there are no surrounding unvisited squares the knight can step to which do not contain the digit 0 or 2.
a(4) = 12 as on a base-4 numbered board the knight steps to squares 22_4 = 10, 3_4 = 3, 12_4 = 6, 33_4 = 15, 2_4 = 2, 11_4 = 5, 20_4 = 8, 111_4 = 21, 220_4 = 40, 13_4 = 7, 222_4 = 42, 103_4 = 19. The knight is then trapped as no unvisited square containing only the digit 2 is one knight step away.
See the linked images for other examples.

Rémy Sigrist, Table of n, a(n) for n = 2..3500
Scott R. Shannon, Image of the path for a(4) = 12. In this and other images the starting square is highlighted in white, the visited squares, numbered in base n, in yellow, the final square in red, while the path is colored across the spectrum to show the relative step ordering.
Scott R. Shannon, Image of the path for a(8) = 35.
Scott R. Shannon, Image of the path for a(10) = 56
Scott R. Shannon, Image of the path for a(16) = 156.
Scott R. Shannon, Image of the path for a(24) = 424.
Scott R. Shannon, Image of the path for a(36) = 144.
Rémy Sigrist, PARI program for A343530

Cf. A344325, A343563, A326918, A328894, A326916, A316667.

PARI
```
See Links section.
```

a(n) = 2015 for any n >= 2979. - Rémy Sigrist, Jun 16 2021

More terms from Rémy Sigrist, Jun 16 2021

A323810 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

Original entry on oeis.org

1, 8, 6, 2, 12, 9, 4, 3, 13, 7, 5, 10, 26, 18, 11, 30, 24, 16, 38, 31, 22, 17, 25, 20, 28, 34, 14, 21, 43, 33, 27, 19, 15, 35, 42, 32, 23, 29, 39, 47, 56, 69, 37, 48, 40, 51, 60, 70, 57, 67, 81, 46, 58, 49, 41, 52, 44, 55, 64, 36, 65, 53, 45, 76, 63, 54, 66, 103, 88, 74, 61

Offset: 1

Daniël Karssen, Jan 28 2019

nonn

Board is numbered as follows:
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
Coincides with A316588 for the first 2402 terms. - Daniël Karssen, Jan 30 2019

Daniël Karssen, Table of n, a(n) for n = 1..100000
Daniël Karssen, Figure showing the first 1e5 steps of the sequence

The sequences involved in this set of related sequences are A316588, A316328, A316334, A316667, A323808, A323809, A323810, and A323811.

A323811 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

Original entry on oeis.org

0, 7, 5, 1, 11, 8, 3, 2, 12, 6, 4, 9, 25, 17, 10, 29, 23, 15, 37, 30, 21, 16, 24, 19, 27, 33, 13, 20, 42, 32, 26, 18, 14, 34, 41, 31, 22, 28, 38, 46, 55, 68, 36, 47, 39, 50, 59, 69, 56, 66, 80, 45, 57, 48, 40, 51, 43, 54, 63, 35, 64, 52, 44, 75, 62, 53, 65, 102, 87, 73, 60

Offset: 0

Daniël Karssen, Jan 28 2019

nonn

Board is numbered as follows:
   0  1  3  6 12 17  .
   2  4  7 13 18  .
   5 10 14 19  .
  11 15 20  .
  16 21  .
  22  .
  .
Coincides with A316334 for the first 2402 terms.

Daniël Karssen, Table of n, a(n) for n = 0..99999
Daniël Karssen, Figure showing the first 1e5 steps of the sequence

The sequences involved in this set of related sequences are A316588, A316328, A316334, A316667, A323808, A323809, A323810, and A323811.

A328894 a(n) is the number of steps before being trapped for a knight starting on square n on a single-digit square-spiral numbered board and where the knight moves to the smallest numbered unvisited square; the minimum distance from the origin is used if the square numbers are equal; the smallest spiral number ordering is used if the distances are equal.

Original entry on oeis.org

1069, 884, 995, 884, 885, 988, 885, 943, 549, 1070, 942, 548, 881, 951, 987, 886, 661, 601, 1123, 1313, 1034, 1070, 1101, 1070, 1930, 943, 655, 882, 1930, 943, 1471, 992, 583, 884, 806, 704, 1062, 1098, 1096, 1129, 1174, 723, 438, 1102, 854

Offset: 1

Scott R. Shannon, Oct 29 2019

nonn

This is the number of completed steps before being trapped for a knight starting on a square with square spiral number n for a knight with step rules given in A326918. We use the standard square spiral number of A316667 to define the start square, as opposed to its single-digit board value, as it is a unique value for each square on the board.
Unlike board numbering methods which have a unique smallest value at the origin, which causes the knight to immediately move toward it when starting from any other square, the single-digit numbering method has multiple small values distributed over the board. Therefore when starting from an arbitrary square the knight may move in any direction, toward the smallest valued neighboring square one knight leap away. Only when two or more such squares exist with the same number does the origin start to act as the square of attraction. This means some knight paths will meander well away from the origin and can become trapped before ever approaching it.
For starting squares n from 1 to 10^6 the longest path before being trapped is a(435525) = 2865. The smallest path to being trapped is a(42329) = 109. The path which ends on the square with the largest standard square spiral number is a(31223), which ends on square 47863. The first path which ends on the square with the smallest standard spiral number is a(138), which ends on square 4. This square is adjacent to the origin, but it is curious that the three squares with smaller spiral numbers, 1,2,3, do not act as the end square for any of the starting squares studied.

			a(1) = 1069. See A326918.
The squares are numbered using single digits of the spiral number ordering as:
                                .
                                .
    2---2---2---1---2---0---2   2
    |                       |   |
    3   1---2---1---1---1   9   3
    |   |               |   |   |
    2   3   4---3---2   0   1   1
    |   |   |       |   |   |   |
    4   1   5   0---1   1   8   3
    |   |   |           |   |   |
    2   4   6---7---8---9   1   0
    |   |                   |   |
    5   1---5---1---6---1---7   3
    |                           |
    2---6---2---7---2---8---2---9
If the knight has a choice of two or more squares in this spiral with the same number which also have the same distance from the origin, then the square with the minimum standard spiral number, as shown in A316667, is chosen.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Eric Angelini, Kneil's Knumberphile Knight, Cinquante signes, May 04 2019.
Eric Angelini, Kneil's Knumberphile Knight, Cinquante signes, May 04 2019. [Cached copy, pdf file, with permission]
Scott R. Shannon, Path for starting square n = 435525. This is trapped after 2865 steps, the longest found path. In this and other images a green square marks the starting square, an orange square the 0-numbered origin square, a red square the ending square, and blue squares mark the eight blocking squares for the end square.
Scott R. Shannon, Path for starting square n = 42329. This is trapped after 109 steps, the shortest found path. This is an example of a path starting and being trapped without approaching the origin. Note that the start square also acts as one of the eight blocking squares.
Scott R. Shannon, Path for starting square n = 31223. This is trapped on the square with standard spiral number 47863, the largest found value. The path also starts and ends without approaching the origin.
Scott R. Shannon, Path for starting square n = 138. This is trapped on the square with standard spiral number 4, the smallest value found.
N. J. A. Sloane and Brady Haran, The Trapped Knight, Numberphile video (2019).

Cf. A316667, A326413, A326916, A174344, A274923, A296030.

A335844 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the lowest spiral number.

Original entry on oeis.org

1, 10, 3, 6, 17, 4, 7, 2, 5, 8, 11, 14, 29, 86, 27, 12, 31, 94, 61, 16, 19, 22, 41, 106, 67, 18, 37, 62, 139, 98, 191, 142, 97, 34, 13, 58, 89, 178, 127, 52, 83, 26, 47, 118, 163, 76, 23, 20, 43, 70, 109, 74, 71, 44, 73, 158, 113, 214, 157, 274, 271, 212, 277, 346, 211

Offset: 1

Scott R. Shannon, Jun 26 2020

This sequences gives the numbers of the squares visited by a knight moving on a square-spiral numbered board, as described in A316667, where at each step the knight goes to the neighbor one knight-leap away which contains the number with the fewest divisors. If two or more neighbors exist with the same fewest number of divisors then the square with the lowest spiral number is chosen.
The sequence is finite. After 528 steps the square with number 33 is visited, after which all neighboring squares have been visited.
Due to the knight's preference for squares with the fewest divisors the knight will leap to a prime numbered square when possible, and the lowest prime if two or more unvisited primes are within one knight-leap. Therefore this sequence matches A330008 for the first 13 terms, but on the 13th step the square with number 86 is chosen as no primes are available and 86 has only four divisors, while A330008 chooses 32, the smallest available number, but which has six divisors.
Of the 528 visited squares 198 contain prime numbers while 330 contain composites. The largest visited square is a(410) = 3656.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square for the knight.
a(2) = 10. The eight unvisited squares the knight can leap to from a(1) are numbered 10,12,14,16,18,20,22,24. Of these 10,14,22 have the minimum four divisors, and of those 10 is the smallest.

Scott R. Shannon, Table of n, a(n) for n = 1..529
Scott R. Shannon, Image showing the 528 steps of the knight's path. A green dot marks the starting 1 square and a red dot the final square with number 33. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 21.

Cf. A316667, A330008, A329520, A326922, A328928, A328929.

A335856 Squares visited by a chess king on a spirally numbered infinite board where the king moves to the adjacent unvisited square containing the lowest prime number. If no such square is available it chooses the lowest-numbered adjacent unvisited square.

Original entry on oeis.org

1, 2, 3, 11, 29, 13, 31, 59, 32, 14, 4, 5, 17, 37, 67, 103, 149, 104, 66, 38, 18, 19, 7, 23, 47, 79, 48, 24, 8, 6, 20, 41, 71, 43, 73, 109, 72, 42, 21, 22, 44, 45, 46, 76, 75, 113, 74, 112, 110, 111, 157, 211, 271, 209, 269, 337, 267, 205, 151, 107, 69, 39, 40, 68, 105, 106, 70, 108

Offset: 1

Scott R. Shannon, Jun 27 2020

This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step moves to the adjacent unvisited square containing the lowest prime number. If no adjacent unvisited square contains a prime number then the square with the lowest spiral number is chosen. Note that if the king simply moves to the lowest unvisited number the sequence will be infinite as the king will just follow the square spiral path.
The sequence is finite. After 719 steps the square with number 437 is visited, after which all adjacent neighboring squares have been visited.
Of the 719 visited squares 165 contain prime numbers while 554 contain composites. As the odd numbers are diagonally adjacent in the square spiral the king's path will contain many diagonal steps, often taking numerous diagonal steps is succession - see the attached link image.
The largest visited square is a(709) = 1367. The lowest unvisited square is 33.
The 719 steps until self-trapping occurs are significantly larger than the expected average of 210 moves to self-trapping for a random walk of the king on an infinite chessboard. See the link to the probability density graphs in A323562. - Hugo Pfoertner, Jul 19 2020
When the grid points are labeled starting with 0 at the origin, the king gets trapped after 171 moves at (3,0), after going as far as (10,-11) to the south-east and (-8,7) and (-5,8) to the north-east, see A383183. - M. F. Hasler, May 13 2025

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square for the king.
a(2) = 2. The four unvisited squares around a(1) the king can move which contain prime numbers are 2,3,5,7. Of those 2 is the lowest.
a(4) = 11. The two unvisited squares around a(3) = 3 the king can move to which contain prime numbers are 11 and 13. Of those 11 is the lowest.
a(9) = 32. There are no unvisited squares around a(8) = 59 which contain prime numbers. The seven other unvisited squares are numbered 32,33,58,60,93,94,95. Of those 32 is the lowest.

M. F. Hasler, Table of n, a(n) for n = 1..720 (complete sequence), May 13 2025
Scott R. Shannon, Image showing the 719 steps of the path. A green dot marks the starting square and a red dot the final square with number 437. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dot marks the smallest unvisited square with number 33.

Cf. A333714, A333713, A335816, A336038, A336092, A316667.
Cf. A174344, A274923, A272763, A323561, A323562.
Cf. A000040 (the primes), A010051 (characteristic function of the primes).

from sympy import isprime # or use A010051
def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real)
    else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y)
def A335856(n, moves=(1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j)):
    if not hasattr(A:=A335856, 'terms'): A.terms=[1]; A.pos=0
    while len(A.terms) < n:
        try: move = min((1-isprime(s), s, z) for d in moves if
                        (s := square_number(z := A.pos+d)+1)not in A.terms)
        except ValueError:
            raise IndexError(f"Sequence has only {len(A.terms)} terms")
        A.terms.append(move[1]); A.pos = move[2]
    return A.terms[n-1]
A335856(999) # gives IndexError: Sequence has only 720 terms
A335856.terms # shows all 720 terms; append [:N] to see only N terms
# M. F. Hasler, May 13 2025

Name edited by Peter Munn, May 11 2025

More terms (complete sequence) from M. F. Hasler, May 13 2025

A336092 Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the largest spiral number.

Original entry on oeis.org

1, 7, 23, 47, 79, 49, 25, 9, 11, 29, 53, 87, 127, 177, 233, 299, 373, 454, 543, 641, 746, 859, 979, 1109, 1247, 1393, 1249, 1111, 983, 863, 751, 647, 753, 866, 865, 985, 1115, 1253, 1399, 1553, 1714, 1883, 2059, 2243, 2437, 2638, 2846, 3063, 3287, 3061, 2843, 2633, 2841, 3057, 3281, 3513, 3755

Offset: 1

Scott R. Shannon, Jul 08 2020

This sequences gives the numbers of the squares visited by a chess king moving on a square-spiral numbered board where the king starts on the 1 numbered square and at each step moves to an adjacent unvisited square, out of the eight adjacent neighboring squares, which contains the number with the fewest divisors. If two or more adjacent squares exist with the same fewest number of divisors then the square with the largest spiral number is chosen. Note that if the king simply moves to the largest available number the sequence will be infinite as the king will step along the south-east diagonal from square 1 forever.
The sequence is finite. After 21276 steps the square with spiral number 281747427 is visited, after which all adjacent neighboring squares have been visited. The end square is extremely far from the starting square, approximately 8860 units away, as the king is drawn generally outward due to its preference for the largest numbered square when the divisor counts are tied - see the link image. This end square spiral number is currently the largest for any square spiral single-visit trapped knight or trapped king path in the OEIS.
Due to the king's preference for squares with the fewest divisors it will move to a prime numbered square when possible, and the lowest prime if two or more unvisited primes are in adjacent squares. Of the 21276 visited squares 4363 contain prime numbers while 16913 contain composites. The largest visited square is a(21208) = 282486458.

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(1) = 1, the starting square for the king.
a(2) = 7. The eight unvisited squares around a(1) the king can move to are numbered 2,3,4,5,6,7,8,9. Of these 2,3,5,7 have the minimum two divisors, and of those 7 is the largest.
a(3) = 23. The seven unvisited squares around a(2) the king can move to are numbered 6,8,19,20,21,22,23. Of these 19 and 23 have the minimum two divisors, and of those 23 is the largest.

Scott R. Shannon, Image showing the 21276 steps of the king's path. A green dot, far lowest left, marks the starting 1 square and a red dot, far upper right, the final square with number 281747427. The red dot is surrounded by eight blue dots to show the occupied neighboring squares. A yellow dots marks the smallest unvisited square with number 2. This is a high resolution image and may need to be downloaded to be viewed correctly.

Cf. A335816 (choose lowest number in case of tie), A333713, A333714, A316667, A330008, A329520, A326922.

A358150 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the smallest numbered unvisited square and where the square number is more than the number of currently visited squares.

Original entry on oeis.org

1, 10, 3, 6, 9, 12, 15, 18, 35, 14, 11, 24, 27, 48, 23, 20, 39, 36, 61, 32, 29, 52, 25, 28, 51, 80, 47, 76, 43, 70, 105, 38, 63, 34, 59, 56, 87, 126, 53, 84, 49, 78, 45, 74, 71, 106, 67, 64, 97, 60, 93, 90, 55, 58, 89, 92, 131, 88, 127, 174, 83, 120, 79, 116, 75, 72, 107, 68, 103, 100, 141

Offset: 1

Scott R. Shannon, Nov 01 2022

This sequence is finite: after 15767 squares have been visited the square with number 15813 is reached after which all eight neighboring squares the knight could move to have already been visited. See the linked image. The largest visited square is a(15525) = 19363, while numerous smaller numbered squares are never visited, e.g., 2, 4, 5, 7, 8, 13, 16, 17, 19, ... .

			The board is numbered with the square spiral:
.
  17--16--15--14--13   .
   |               |   .
  18   5---4---3  12   29
   |   |       |   |   |
  19   6   1---2  11   28
   |   |           |   |
  20   7---8---9--10   27
   |                   |
  21--22--23--24--25--26
.
a(6) = 12 as after the knight moves to the square containing 9 the available unvisited squares are 4, 12, 22, 26, 28, 46, 48. Of these 4 is the smallest but as we have already visited five squares that cannot be chosen. Of the remaining squares greater than five the smallest unvisited square is 12. This is the first term to differ from A316667.

Scott R. Shannon, Image showing the knight's path on the square spiral. The starting 1 square is shown as a green dot while the final square numbered 15813, near the middle of the top edge, is shown as a red dot. Also shown as blue dots are the eight occupied squares around the final square.

Cf. A316667, A326918, A326922, A316588.

A383185 Number of the square visited by a king moving on a spirally numbered board always to the lowest available unvisited square, when a wall delimiting the spiral must be crossed on each move.

Original entry on oeis.org

0, 3, 13, 2, 10, 1, 7, 21, 6, 18, 4, 14, 32, 12, 28, 11, 27, 9, 23, 8, 22, 44, 20, 40, 19, 5, 17, 37, 16, 34, 15, 33, 59, 31, 57, 30, 54, 29, 53, 85, 51, 25, 47, 24, 46, 76, 45, 75, 43, 73, 42, 70, 41, 69, 39, 67, 38, 66, 36, 62, 35, 61, 95, 60, 94, 58, 92, 56, 88, 55, 87, 127, 86, 52, 26

Offset: 0

M. F. Hasler, May 12 2025

The board is numbered following a square spiral starting with 0 at the origin (where the king is at n = 0) and delimited by a wall that must be crossed on each move:
.
  16  15  14  13  12 | .
     ,-----------.   | .
  17 | 4   3   2 |11 | .
     |   ,---    |   | .
  18 | 5 | 0   1 |10 | .
     |   '-------'   | .
  19 | 6   7   8   9 | .
     `---------------' .
  20  21  22  23  24  25
.
A line drawn from the center of the starting square to the center of the ending square must pass through a wall on each move. A move that would just touch a wall without passing through the wall (e.g., 0 to 2) is not permissible. Equivalently, the king can't move from a square labeled k to a square labeled k +- 1 or k +- 2, i.e., |a(n)-a(n+1)| > 2 for all n.
This sequence is a permutation of the nonnegative integers, see A383186 for the inverse permutation. The king's walk indeed fills the 2D grid with an initial segment S0 of 24 moves, followed by rings R(r), r >= 1, which consist of three shells S1(r), S2(r) and S3(r), each of which corresponds to a tour around the center. Each ring R(r) starts with the move number n = 48 r^2 - 16 r + 1 = (25, 147, 365, ...) to the square at position P(r) = (2-3r, 3r-3) = ((-1,0), (-4,3), (-7,6), ...), and contains a perfectly well defined sequence of 96 r + 26 grid points following a precise sequence of pattern given in full detail on the wiki page provided in the links section.

			For n = 1, a(1) = 3 because moving from 0 to 1 or 2 does not pass through a wall.

M. F. Hasler, Table of n, a(n) for n = 0..1000
M. F. Hasler, A383185: Details about the grid filling king's walk, OEIS wiki, May 15, 2025.
M. F. Hasler, Path plot with annotations, May 15, 2025.
OEIS index to sequences related to permutations of the integers.

Cf. A375925 (the same with indices and numbers of squares starting at 1).
Cf. A383186 (inverse permutation).
Cf. A316328 (knight's path), A033638, A316667 (trapped knight), A336038 (trapped king), A335856 (trapped king preferably moving to prime numbers).

def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real)
    else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y)
def A383185(n):
    if not hasattr(A:=A383185, 'terms'): A.terms=[0]; A.pos=0; A.path=[0]
    while len(A.terms) <= n:
        s,d = min((s,d) for d in (1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j) if
            abs((s:=square_number(A.pos+d))-A.terms[-1]) > 2 and s not in A.terms)
        A.terms.append(s); A.pos += d; A.path.append(A.pos)
    return A.terms[n]
import matplotlib.pyplot as plt # this and below to plot the trajectory
plt.plot([z.real for z in A383185.path], [z.imag for z in A383185.path])
plt.show()

cp's OEIS Frontend

A336038 Squares visited by a chess king on a square-spiral numbered board and stepping to the lowest unvisited adjacent square, where each step is not in the same direction as the previous step.

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A323810 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

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A323811 Squares visited by a knight on a diagonally numbered board and moving to the lowest available unvisited square at each step and if no unvisited squares are available move one step back.

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A335844 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the lowest spiral number.

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A335856 Squares visited by a chess king on a spirally numbered infinite board where the king moves to the adjacent unvisited square containing the lowest prime number. If no such square is available it chooses the lowest-numbered adjacent unvisited square.

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A336092 Squares visited by a chess king moving on a square-spiral numbered board where the king moves to the adjacent unvisited square containing the spiral number with the fewest divisors. In case of a tie it chooses the square with the largest spiral number.

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A358150 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to the smallest numbered unvisited square and where the square number is more than the number of currently visited squares.

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A383185 Number of the square visited by a king moving on a spirally numbered board always to the lowest available unvisited square, when a wall delimiting the spiral must be crossed on each move.

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