A220098 -id:A220098 - cp's OEIS frontend

A220102 Permutation of natural numbers arising from applying the walk of square spiral (e.g. A214526) to the data of double square spiral (defined in A220098).

1, 2, 4, 6, 8, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 50, 52, 54, 56

Offset: 1

Alex Ratushnyak, Dec 04 2012

nonn

Cf. A214526, A220098, A217010, A217011, A217012, A217013, A217014, A217015.

#include 
#define SIZE 20
int grid[SIZE][SIZE];
int direction[] = {0, -1,  1, 0, 0, 1, -1, 0};
main() {
  int i, j, x1, y1, x2, y2, stepSize;
  int direction1pos=0, direction2pos=4, val;
  x1 = y1 = x2 = y2 = SIZE/2;
  for (val=grid[y1][x1]=1, stepSize=0; ; ++stepSize) {
    if (x1<1 || x1>=SIZE-1 || x2<1 || x2>=SIZE-1) break;
    if (y1<1 || y1>=SIZE-1 || y2<1 || y2>=SIZE-1) break;
    for (i=stepSize|1; i; ++val,--i) {
      x1 += direction[direction1pos  ];
      y1 += direction[direction1pos+1];
      x2 += direction[direction2pos  ];
      y2 += direction[direction2pos+1];
      grid[y1][x1] = val*2;
      grid[y2][x2] = val*2+1;
    }
    direction1pos = (direction1pos+2) & 7;
    direction2pos = (direction2pos+2) & 7;
  }
  direction1pos=0;
  x1 = y1 = SIZE/2;
  for (stepSize=2; ; ++stepSize) {
    for (i=stepSize/2; i; --i) {
      if (grid[y1][x1]==0) return;
      printf("%d, ",grid[y1][x1]);
      x1 += direction[direction1pos  ];
      y1 += direction[direction1pos+1];
    }
    direction1pos = (direction1pos+2) & 7;
  }
}

A332837 Squares visited by a knight moving on a double spiral numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 10, 5, 2, 8, 7, 4, 3, 9, 6, 12, 18, 33, 39, 20, 11, 32, 19, 13, 22, 28, 15, 21, 38, 61, 30, 17, 42, 25, 31, 16, 43, 24, 51, 76, 26, 45, 70, 37, 14, 29, 23, 40, 34, 57, 86, 49, 55, 84, 78, 53, 47, 72, 107, 41, 35, 56, 27, 44, 71, 36, 59, 88, 127, 80, 115

Offset: 1

Scott R. Shannon, Feb 26 2020

This sequence uses a double spiral of numbers to enumerate the squares on the board. The knight starts on the square with number 1. At each step the knight goes to an unvisited square with the smallest number.
The sequence is finite. After 2958 steps the square with number 2796 is visited, after which all neighboring squares have been visited.
The lowest unvisited square during the walk is square number 2011.

			The squares are numbered using the double spiral numbering shown below:
.
  --48--46--44--42--40--38--36
                             |
    27--25--23--21--19--17  34
     |                   |   |
    29  10---8---6---4  15  32
     |   |           |   |   |
    31  12   3---1---2  13  30
     |   |   |           |   |
    33  14   5---7---9--11  28
     |   |                   |
    35  16--18--20--22--24--26
     |
    37--39--41--43--45--47--49--

Scott R. Shannon, Table of n, a(n) for n = 1..2959
Scott R. Shannon, Image showing the 2958 steps of the knights' path. The green dot is the starting square and the red dot the final square. Blue dots show the eight occupied squares surrounding the final square. The lowest unvisited square is the yellow dot.

Cf. A220098, A316667, A329022, A332980 (quadruple spiral).

A332980 Squares visited by a knight moving on a quadruple spiral numbered board and moving to the lowest available unvisited square at each step.

Original entry on oeis.org

1, 10, 7, 2, 8, 3, 9, 4, 6, 5, 15, 12, 38, 11, 14, 21, 32, 20, 17, 28, 39, 13, 16, 19, 30, 18, 33, 44, 56, 37, 22, 34, 23, 35, 24, 36, 25, 52, 71, 29, 40, 59, 47, 66, 31, 42, 54, 77, 89, 27, 62, 43, 55, 74, 86, 117, 70, 51, 94, 67, 48, 60, 41, 26, 65, 92, 49

Offset: 1

Scott R. Shannon, Mar 04 2020

This sequence uses a quadruple spiral of numbers to enumerate the squares on the board. The knight starts on the square with number 1. At each step the knight goes to an unvisited square with the smallest number.
The sequence is finite. After 1837 steps the square with number 1748 is visited, after which all neighboring squares have been visited.
The lowest unvisited square during the walk is square number 1211.

			The squares are numbered using the quadruple spiral numbering shown below:
                             |
  --49--45--41--37--33--29  48
                         |   |
    26--22--18--14--10  25  44
     |               |   |   |
    30  11---7---3   6  21  40
     |   |       |   |   |   |
    34  15   4-- 1---2  17  36
     |   |   |   |       |   |
    38  19   8   5---9--13  32
     |   |   |               |
    42  23  12--16--20--24--28
     |   |
    46  27--31--35--39--43--47--
     |

Scott R. Shannon, Table of n, a(n) for n = 1..1838
Scott R. Shannon, Image showing the 1837 steps of the knights' path. The green dot is the starting square and the red dot the final square. Blue dots show the eight occupied squares surrounding the final square. The lowest unvisited square is the yellow dot.

Cf. A220098, A316667, A329022, A332837 (double spiral).

A220099 Sum of the eight nearest neighbors of n in a double spiral with positive integers and 1 at the center.

Original entry on oeis.org

44, 66, 63, 94, 91, 81, 80, 101, 100, 158, 154, 133, 131, 153, 151, 222, 218, 185, 183, 190, 186, 206, 202, 237, 235, 318, 314, 269, 267, 270, 266, 286, 282, 321, 319, 414, 410, 353, 351, 350, 346, 366, 362, 382, 378, 398, 394, 437, 435, 542, 538, 469, 467, 462, 458

Offset: 1

Alex Ratushnyak, Dec 04 2012

Double spiral begins:
.
  82---84---86---88---90---92---94---96---98
   |
  80   51---53---55---57---59---61---63---65
   |    |                                  |
  78   49   26---28---30---32---34---36   67
   |    |    |                        |    |
  76   47   24   11---13---15---17   38   69
   |    |    |    |              |    |    |
  74   45   22    9    2----4   19   40   71
   |    |    |    |    |    |    |    |    |
  72   43   20    7    1    6   21   42   73
   |    |    |    |    |    |    |    |    |
  70   41   18    5----3    8   23   44   75
   |    |    |              |    |    |    |
  68   39   16---14---12---10   25   46   77
   |    |                        |    |    |
  66   37---35---33---31---29---27   48   79
   |                                  |    |
  64---62---60---58---56---54---52---50   81
                                           |
  99---97---95---93---91---89---87---85---83

Cf. A220098, A220100.

A220100 Sum of the four nearest neighbors of n in a double spiral with positive integers and 1 at the center.

Original entry on oeis.org

18, 27, 26, 42, 42, 34, 35, 42, 42, 74, 74, 58, 58, 66, 66, 106, 106, 82, 82, 90, 90, 98, 98, 106, 106, 154, 154, 122, 122, 130, 130, 138, 138, 146, 146, 202, 202, 162, 162, 170, 170, 178, 178, 186, 186, 194, 194, 202, 202, 266, 266, 218, 218, 226, 226, 234, 234, 242

Offset: 1

Alex Ratushnyak, Dec 04 2012

Double spiral begins:
.
  82---84---86---88---90---92---94---96---98
   |
  80   51---53---55---57---59---61---63---65
   |    |                                  |
  78   49   26---28---30---32---34---36   67
   |    |    |                        |    |
  76   47   24   11---13---15---17   38   69
   |    |    |    |              |    |    |
  74   45   22    9    2----4   19   40   71
   |    |    |    |    |    |    |    |    |
  72   43   20    7    1    6   21   42   73
   |    |    |    |    |    |    |    |    |
  70   41   18    5----3    8   23   44   75
   |    |    |              |    |    |    |
  68   39   16---14---12---10   25   46   77
   |    |                        |    |    |
  66   37---35---33---31---29---27   48   79
   |                                  |    |
  64---62---60---58---56---54---52---50   81
                                           |
  99---97---95---93---91---89---87---85---83

			The four nearest neighbors of 8 are 3, 6, 23, 10; their sum is a(8)=42.
The four nearest neighbors of 9 are 22, 11, 2, 7; their sum is a(9)=42.

Cf. A220098, A220099.

cp's OEIS Frontend

A220102 Permutation of natural numbers arising from applying the walk of square spiral (e.g. A214526) to the data of double square spiral (defined in A220098).

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A332837 Squares visited by a knight moving on a double spiral numbered board and moving to the lowest available unvisited square at each step.

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A332980 Squares visited by a knight moving on a quadruple spiral numbered board and moving to the lowest available unvisited square at each step.

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A220099 Sum of the eight nearest neighbors of n in a double spiral with positive integers and 1 at the center.

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A220100 Sum of the four nearest neighbors of n in a double spiral with positive integers and 1 at the center.

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