A328929 -id:A328929 - cp's OEIS frontend

A333713 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 most divisors. In case of a tie it chooses the square with the lowest spiral number.

1, 6, 18, 40, 70, 108, 72, 42, 20, 21, 44, 45, 75, 114, 160, 216, 280, 350, 351, 352, 432, 520, 616, 720, 832, 952, 1080, 1216, 1360, 1512, 1672, 1840, 2016, 2200, 2392, 2592, 2800, 3016, 3240, 3472, 3710, 3956, 4212, 4476, 4746, 5024, 5310, 5022, 4743, 4472, 4473, 4209, 4208, 3952, 3705

Offset: 1

Scott R. Shannon, Jul 02 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 most divisors. If two or more adjacent squares exist with the same highest number of divisors then the square with the lowest spiral number is chosen. Note that if the king simply moves to the highest 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 1784 steps the square with number 1478 is visited, after which all adjacent neighboring squares have been visited.
Due to the king's preference for squares with the most divisors it will avoid prime numbers unless no other choice exists. Of the 1784 visited squares only 27 contain prime numbers while 1757 contain composites. As even numbers >= 6 will always contain 4 or more divisors the king will tend to visit more even numbers than odd numbers; in the 1784 visited squares 1289 contain an even number while 495 contain an odd number. As the even numbers are diagonally adjacent in the square spiral the king's path will be dominated by diagonal steps, often taking numerous diagonal steps is succession - see the attached link image.
The largest visited square is a(390) = 17664. The lowest unvisited square is 2.

			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) = 6. The eight unvisited squares around a(1) the king can move to are numbered 2,3,4,5,6,7,8,9. Of these 6 and 8 both have the maximum four divisors, and of those 6 is the smallest.
a(3) = 18. The seven unvisited squares around a(2) = 6 the king can move to are numbered 4,5,18,19,20,7,8. Of these 18 and 20 have the maximum six divisors, and of those 18 is the smallest.
a(603) = 821. This is the first prime number visited; a(602) = 939 has square 821 as the sole unvisited adjacent neighbor.

Scott R. Shannon, Table of n, a(n) for n = 1..1785
Scott R. Shannon, Image showing the 1784 steps of the king's path. A green dot marks the starting 1 square and a red dot the final square with number 1478. 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.

Cf. A333714 (choose highest spiral number in case of tie), A335816, A316667, A330008, A329520, A326922, A328928, A328929.

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.

A358278 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 is on a different square ring of numbers than the current square.

Original entry on oeis.org

1, 10, 3, 16, 33, 4, 11, 8, 19, 38, 5, 14, 29, 2, 13, 28, 9, 12, 27, 24, 7, 18, 35, 60, 15, 6, 17, 34, 59, 30, 53, 26, 79, 46, 21, 40, 67, 36, 61, 32, 55, 86, 51, 48, 23, 44, 71, 20, 39, 66, 99, 62, 37, 68, 41, 22, 43, 70, 105, 148, 65, 98, 139, 94, 31, 54, 85, 50, 25, 52, 49, 78, 45, 74

Offset: 1

Scott R. Shannon and Eric Angelini, Nov 08 2022

This sequence is finite: after 1455 squares have been visited the square with number 1345 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(1374) = 1996 while the smallest unvisited square is 1024.

			The board is numbered using a square spiral. The square rings of numbers are shown below:
.
    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
                         |
   -44--45--46--47--48--49
.
a(4) = 16 as after the knight moves to the square containing a(3) = 3 the available unvisited squares are 6, 8, 16, 28, 30, 32, 34. Of these 6 and 8 are the smallest but both of them lie on the first square ring of numbers, the same as the current number 3. Of the remaining squares the smallest unvisited square is 16. 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 1345, near the middle of the bottom edge, is shown as a red dot. Also shown as blue dots are the eight occupied squares around the final square.

Cf. A316667, A358150, A328929, A328909, A174344, A274923.

A362027 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to a previously unvisited square with a number as close as possible to the number of the current square. If two such squares exist the smaller numbered square is chosen.

Original entry on oeis.org

1, 10, 3, 6, 9, 12, 15, 18, 7, 4, 11, 8, 5, 2, 13, 28, 25, 46, 21, 40, 17, 34, 59, 56, 29, 32, 55, 58, 33, 30, 53, 26, 47, 22, 19, 16, 37, 62, 95, 136, 91, 130, 87, 52, 49, 24, 27, 48, 51, 80, 83, 120, 123, 84, 81, 118, 77, 44, 41, 68, 103, 100, 63, 66, 39, 36, 61, 94, 57, 88, 127, 174, 229, 170

Offset: 1

Scott R. Shannon, Apr 05 2023

This sequence is finite: after 130 squares have been visited the square with number 50 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(117) = 247 while the smallest unvisited square is 20.

			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 12, where |12 - 9| = 3, is the closest number to 9. This is the first term to differ from A316667.

Scott R. Shannon, Table of n, a(n) for n = 1..130.
Scott R. Shannon, Image showing the 129 steps of the knight's path. The first and last squares are highlighted in green and red respectively while the eight squares blocking the final square are surrounded in blue.

Cf. A316667, A358278, A358150, A328929, A328909, A174344, A274923.

A377015 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to a square which has been previously visited the fewest number of times. If two or more such squares exist the smallest numbered square is chosen.

Original entry on oeis.org

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, 39, 66, 63, 96, 59, 56, 87, 52, 49, 78, 115, 74, 71, 106, 149, 102, 99, 140, 61, 94, 31, 54, 85, 50

Offset: 1

Scott R. Shannon, Nov 09 2024

Unlike similar sequences, e.g. A316667, A362027, A326922, in this variation the knight is never trapped as it can always move to the square which has been previously visited the fewest times, or if two or more surrounding squares exist with the same smallest previous visit count, then it can move to the smallest numbered square of these options.
The first 2016 terms are the same as A316667. In that sequence the path now ends, but here, as the knight is now surrounded by eight squares that have all been visited once, it now chooses the smallest numbered available square, 1733 in this case. This eventually leads it back toward the origin where it revisits the 1 starting square at a(2039). From here it once again chooses the surrounding square with the fewest previous visits, so it begins a new path, but it will avoid the path it previously took back to the origin since those squares will have two previous visits.
The above pattern repeats, causing the knight to go on various excursions of generally increasing length before it revisits the origin - see A377928 for the indices where a(n) = 1. The knight eventually tours paths of increasing complexity, in general moving along distorted loops which are defined by the previous visit count, with straight path lines between these loops. See the attached images.
Interestingly some of the paths between origin visits are identical; for example the path between the 2nd and 3rd visits to the origin is the same as that between the 32nd and 33rd visits. Likewise those between the 5th-6th and 23rd-24th, and 37th-38th and 40th-41st are the same.

			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 default starting square.
a(2) = 10 as all eight surrounding available squares, 10, 12, 14, 16, 18, 20, 22, 24 have zero previous visits, so it chooses the smallest number of those, namely 10.
a(3) = 3 as there are seven available squares that have zero previous visits, and of those 3 is the smallest number. Note the 1 square is not considered as that has one previous visit which is more than the other seven squares.
a(2017) = 1733 as all eight surrounding available squares have been visit once, so it chooses the smallest number of those, namely 1733. This is the first term to differ from A316667.

Scott R. Shannon, Table of n, a(n) for n = 1..20000
Scott R. Shannon, Image of the first 100000 terms.
Scott R. Shannon, Image of the path between the first and second visit to the origin. In this and similar images, the steps are colored white, grey, red, orange, yellow, green, blue, indigo, and violet across the entire path to show their relative ordering. This image is the same as in A316667 except for the additional final steps, colored violet, back to the origin starting at a(2017). Also show as red circles are the visited squares whose eight available neighbours all have the same previous visit count.
Scott R. Shannon, Image of the path between the second and third visits to the origin.
Scott R. Shannon, Image of the path between the third and fourth visits to the origin. This shows the path beginning to get more complex - the knight travels along different loops of squares during different parts of its walk, moving between loops via generally straight line paths.
Scott R. Shannon, Image of the path between the sixth and seventh visits to the origin.
Scott R. Shannon, Image of the path between the eighth and ninth visits to the origin
Scott R. Shannon, Image of the path between the twelfth and thirteenth visits to the origin.
Scott R. Shannon, Image of the path between the twenty-third and twenty-fourth visits to the origin. This is identical to the path between the fifth and sixth visits.
Scott R. Shannon, Image of the path between the twenty-ninth and thirtieth visits to the origin.

Cf. A377928, A316667, A362027, A343678, A358278, A358150, A328929, A328909.

A357046 Squares visited by a knight moving on a board covered with horizontal dominoes [m|m], m = 0, 1, 2, ... in a diamond-shaped spiral, when the knight always jumps to the unvisited square with the least number on the corresponding domino.

Original entry on oeis.org

0, 11, 14, 1, 4, 13, 10, 3, 18, 7, 2, 5, 22, 9, 28, 31, 60, 15, 32, 29, 52, 25, 8, 27, 12, 53, 26, 23, 6, 17, 34, 59, 30, 87, 126, 51, 24, 45, 20, 39, 16, 33, 58, 55, 86, 125, 50, 47, 76, 21, 40, 67, 36, 61, 94, 57, 54, 85, 176, 129, 56, 93, 138, 187, 92, 137, 96, 35, 38, 19

Offset: 0

M. F. Hasler, Oct 19 2022

The sequence lists the squares visited by the knight by giving their (unique) "square spiral number", as shown, e.g., in A316328 and others. (Listing the labels m of the dominoes would obviously be ambiguous; see EXAMPLE for that sequence.)
The dominoes [m|m], m = 0, 1, 2, ... are placed in a diamond-shaped spiral,
           12  12  28  28      
_        13  13  11  11  27  27   _
     14  14  [2 | 2] 10  10  26  26
_  15  15  [3 | 3] [1 | 1] [9 | 9] 25
_  16  [4 | 4] [0 | 0] [8 | 8] 24  24
  17  17  [5 | 5] [7 | 7] 23  23   
_     18  18  [6 | 6] 22  22      _
        19  19  21  21         
The spiral starts from the origin (where the [0|0] is placed) with one step in direction North-East (where [1|1] is placed), then one in direction North-West (=> [2|2]), then two towards South-West (=> [3|3] and [4|4]) and two towards South-East (=> [5|5] and [6|6]), then three towards North-East, etc. [We chose the counter-clockwise spiral as usual in mathematics, but one would obviously get the same sequence if the spiral of dominoes and the square spiral numbering the positions were chosen in the opposite, clockwise sense.]
The endpoints of the "straight lines" are labeled with the "quarter-squares" A002620, in particular, rightmost and leftmost dominoes of each "shell" are labeled with the odd resp. even square numbers.
The sequence ends at a(2550) where the knight is stuck at position (x, y) = (28, 4) on the domino labeled m = 964.

			The knight hops from the left 0 (= the origin) on the right 1, then on the left 2, then on the right 0, then on the left 3, then on the right 2, etc.
The list of these labels would be 0, 1, 2, 0, 3, 2, 8, 3, 4, 5, 1, 4, 6, 7, 9, 11, 12, 14, 11, 10, 24, 22, 7, 8, 10, 9, 23, 6, 5, 15, 13, 12, 27, 26, 48, 23, ...
As explained in comments, the terms a(n) correspond to the (unique) "square spiral numbers" of these locations (cf. A274641 or A174344 (upside down) or A316328).

Eric Angelini, Spirals for Scott, Blog entry, Oct. 19, 2022.
Eric Angelini, Spirals for Scott, Blog entry, Oct. 19, 2022 [Local copy, with permission].
M. F. Hasler, Plot of the knight's tour A357046, Oct 20 2022.

Cf. A316328, A326924 and A326922 (choose square closest to the origin), A328908 and A328928 (variant using taxicab distance); A328909 and A328929 (variant using sup norm).

Cf. A274641, A174344 (upside down), A268038, A274923 for the square spiral numbering and corresponding (x,y) coordinates.

/* function domino([x,y]) gives the label m on the domino at (x,y); it uses the map DOM to store this label with key x + i*y. */
DOM=Map(); {domino(x)=while(!mapisdefined(DOM, x[1]+I*x[2], &x), my(M=#DOM\2, side=sqrtint(M*4-!!M), pos=sqrtint(M)*I^(side-1)+side\/2%2*I, dir=(1+I)*I^side); for(m=M, M+side\2, mapput(DOM, pos, m); mapput(DOM, pos+1, m); pos+=dir)); x}
{coords(n, m=sqrtint(n), k=m\/2)=if(m<=n-=4*k^2, [n-3*k, -k], n>=0, [-k, k-n], n>=-m, [-k-n, k], [k, 3*k+n])}
{local(U=[]/* used squares */, K=vector(8, i, [(-1)^(i\2)<<(i>4), (-1)^i<<(i<5)])/* knight moves */, pos(x, y)=if(y>=abs(x), 4*y^2-y-x, -x>=abs(y), 4*x^2-x-y, -y>=abs(x), (4*y-3)*y+x, (4*x-3)*x+y), t(x, p=pos(x[1], x[2]))=if(p<=U[1]||setsearch(U, p), oo, [domino(x), p]), nxt(p, x=coords(p))=vecsort(apply(K->t(x+K), K))[1][2]); my(A=List(0)/*list of positions*/); for(n=1, oo, U=setunion(U, [A[n]]); while(#U>1&&U[2]==U[1]+1, U=U[^1]); iferr(listput(A, nxt(A[n])), E, break)); print("Index of last term: ", #A-1); A357046(n)=A[n+1];} \\ same code as A326924 except for norml2 => domino
/* to get the sequence of labels m (cf.example): */
[domino(coords(A357046(n))) | n <- [0..99]]

A361377 Squares visited by a knight moving on a spirally numbered board always to the lowest unvisited coprime square.

Original entry on oeis.org

1, 10, 3, 8, 5, 2, 7, 4, 9, 22, 19, 16, 33, 58, 13, 28, 25, 46, 21, 40, 17, 6, 23, 20, 39, 70, 43, 76, 47, 26, 11, 14, 29, 32, 15, 62, 37, 18, 35, 38, 63, 34, 59, 30, 53, 12, 31, 54, 85, 124, 51, 80, 83, 52, 49, 24, 77, 48, 119, 50, 27, 86, 55, 128, 89, 92

Offset: 1

Jodi Spitz, Mar 09 2023

Many of these sequences (see cross-references) are finite. I've worked this out by hand, but I suspect this sequence is also finite.

The sequence is finite with 156 terms. - Rémy Sigrist, Mar 12 2023

			The spiral board begins:
   .---.---.--33--32--31
                       |
  17--16--15--14--13  30
   |               |   |
  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(9) = 9 and a(10) = 22. For a knight on square 9, the smallest unused square which is both coprime to and a knight's move away from 9 is 22.

Rémy Sigrist, Table of n, a(n) for n = 1..156
Rémy Sigrist, PARI program

Cf. A316667, A326922, A328929, A328928.

PARI
```
See Links section.
```

Data corrected by Rémy Sigrist, Mar 12 2023

A377928 The indices k where A377015(k) = 1.

Original entry on oeis.org

1, 2039, 2703, 30083, 32155, 32437, 86925, 292101, 339137, 430611, 669371, 670563, 727051, 1161819, 1534325, 1541819, 1543011, 2027935, 2718001, 3266661, 3273829, 3730467, 4805861, 4806143, 5534871, 6371063, 7834735, 8926025, 9293575, 9664815, 12629449, 13645059, 13645723, 16510691, 19947389, 19952425, 22519739, 22520381, 24820941, 26657853, 26658495

Offset: 1

Scott R. Shannon, Nov 11 2024

nonn

These are the indices where a knight moving on a square spiral revisits the origin using the path rules given in A377015. See that sequence for further details.

Cf. A377015, A316667, A362027, A343678, A358278, A358150, A328929, A328909.

A364247 Squares visited by the chess king on a spiral-numbered board, where the king moves to the square with the fewest steps to reach 1 using the 3x+1 function. In case of a tie, the king moves to the square with the smallest number.

Original entry on oeis.org

1, 2, 4, 16, 5, 6, 8, 24, 10, 26, 48, 80, 120, 168, 122, 170, 226, 227, 228, 172, 173, 174, 232, 176, 128, 88, 56, 90, 92, 136, 93, 58, 32, 13, 3, 12, 11, 28, 52, 84, 85, 53, 29, 30, 31, 57, 89, 130, 180, 181, 131, 132, 133, 184, 244, 186, 245, 312, 246, 314

Offset: 1

Wagner Martins, Jul 15 2023

The king moves to the square with the fewest steps to reach 1 using the 3x+1 function. The function works as follows: start with the number, and if it is even, divide it by 2. Otherwise, multiply it by 3 and add 1, and repeat the process until you reach 1. If there are two squares with the same number of steps, the king picks the square with the smaller number.
The sequence contains 511 terms; the king gets stuck because all the adjacent squares are already taken.
The last square visited is numbered a(511) = 6619.
The highest-numbered square reached is a(327) = 12853.

			The spiral board:
  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 initial square.
a(2) = 2 because 2 has the fewest steps to reach 1 applying the function {n/2 if n is even, 3n + 1 if n is odd} repeatedly.

Cf. A014682, A006577, A335816, A316667, A330008, A329520, A326922, A328928, A328929.

class Spiral:
    def _init_(self):
        self.spiral = [[1]]
    def increment(self, increment_size):
        if increment_size == 0:  # Recursion stop condition
            return
        size = len(self.spiral)
        count = size ** 2 + 1
        if size % 2 != 0:
            self.spiral.insert(0, [])
            for i in reversed(range(0, size + 1)):
                self.spiral[i].append(count)
                count += 1
            for _ in range(size):
                self.spiral[0].insert(0, count)
                count += 1
        else:
            self.spiral.append([])
            for i in range(0, size + 1):
                self.spiral[i].insert(0, count)
                count += 1
            for _ in range(size):
                self.spiral[-1].append(count)
                count += 1
        self.increment(increment_size - 1)
    def find_position(self, target):
        for i, row in enumerate(self.spiral):
            for j, element in enumerate(row):
                if element == target:
                    return (i, j)
    def find_king_neighbours(self, target):
        i, j = self.find_position(target)
        neighbours_position = (
            (i - 1, j - 1), (i - 1, j), (i - 1, j + 1),
            (i, j - 1), (i, j + 1),
            (i + 1, j - 1), (i + 1, j), (i + 1, j + 1)
        )
        return [self.spiral[i][j] for i, j in neighbours_position]
def steps(x):
    count = 0
    while x != 1:
        if x % 2 == 0:
            x //= 2
        else:
            x = 3 * x + 1
        count += 1
    return count
def min_steps(lst):
    """Find the value with the minimal amount of steps with the 3x+1 function (the smallest in case of tie)"""
    if len(lst) == 0:
        raise ValueError("Empty list")
    min_steps_seen, min_seed = float("inf"), float("inf")
    for n in lst:
        step = steps(n)
        if step < min_steps_seen or step == min_steps_seen and n < min_seed:
            min_steps_seen = step
            min_seed = n
    return min_seed
spiral = Spiral()
sequence = [1]
count = 1
print(count, 1)
while True:
    count += 1
    spiral.increment(2)
    neighbours = spiral.find_king_neighbours(sequence[-1])
    neighbours = [n for n in neighbours if n not in sequence]
    try:
        next_square = min_steps(neighbours)
    except ValueError:
        print("End of the sequence.")
        break
    sequence.append(next_square)
    print(count, sequence[-1])

cp's OEIS Frontend

A333713 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 most divisors. In case of a tie it chooses the square with the lowest spiral number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A358278 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 is on a different square ring of numbers than the current square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A362027 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to a previously unvisited square with a number as close as possible to the number of the current square. If two such squares exist the smaller numbered square is chosen.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A377015 Squares visited by a knight moving on a square-spiral numbered board where the knight moves to a square which has been previously visited the fewest number of times. If two or more such squares exist the smallest numbered square is chosen.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A357046 Squares visited by a knight moving on a board covered with horizontal dominoes [m|m], m = 0, 1, 2, ... in a diamond-shaped spiral, when the knight always jumps to the unvisited square with the least number on the corresponding domino.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A361377 Squares visited by a knight moving on a spirally numbered board always to the lowest unvisited coprime square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A377928 The indices k where A377015(k) = 1.

Views

Author

Keywords

Comments

Crossrefs

A364247 Squares visited by the chess king on a spiral-numbered board, where the king moves to the square with the fewest steps to reach 1 using the 3x+1 function. In case of a tie, the king moves to the square with the smallest number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python