A174344 -id:A174344 - cp's OEIS frontend

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.

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.

A308412 Indices of Gaussian primes on a square spiral.

Original entry on oeis.org

3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 54, 60, 62, 68, 70, 76, 78, 82, 84, 88, 90, 92, 94, 98, 100, 102, 104, 108, 110, 112, 114, 118, 120, 122, 126, 128, 132, 134, 138, 140, 144, 146, 150, 152, 156, 158

Offset: 1

Rémy Sigrist, Jun 01 2019

nonn

These are the numbers k > 0 such that A174344(k) + i*A274923(k) is a Gaussian prime (where i denotes the imaginary unit).
For symmetry reasons, we obtain the same sequence when considering a clockwise or a counterclockwise square spiral, or when initially moving towards any unit direction.
All terms except the first four are even.

			The first terms displayed on the center of a counterclockwise square spiral are:
  y\x|    -5   -4   -3   -2   -1    0   +1   +2   +3   +4   +5
  ---+--------------------------------------------------------
   +5|     *--100----*---98----*----*----*---94----*---92----*
     |     |                                                 |
   +4|   102    *----*----*---62----*---60----*----*----*   90
     |     |    |                                       |    |
   +3|     *    *    *---36----*---34----*---32----*    *    *
     |     |    |    |                             |    |    |
   +2|   104    *   38    *---16----*---14----*   30    *   88
     |     |    |    |    |                   |    |    |    |
   +1|     *   68    *   18    5----*----3   12    *   54    *
     |     |    |    |    |    |         |    |    |    |    |
    0|     *    *   40    *    *    *----*    *   28    *    *
     |     |    |    |    |    |              |    |    |    |
   -1|     *   70    *   20    7----*----9---10    *   52    *
     |     |    |    |    |                        |    |    |
   -2|   108    *   42    *---22----*---24----*---26    *   84
     |     |    |    |                                  |    |
   -3|     *    *    *---44----*---46----*---48----*----*    *
     |     |    |                                            |
    4|   110    *----*----*---76----*---78----*----*----*---82
     |     |
    5|     *--112----*--114----*----*----*--118----*--120----*

Robert Israel, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A308412
Eric Weisstein's World of Mathematics, Gaussian Prime

Cf. A174344, A274923.

SP:= proc(n) option remember; local k;
k:=floor(sqrt(4*n-7)) mod 4;
procname(n-1) -I*exp(I*k*Pi/2)
end proc:
SP(1):= 0:
select(i -> GaussInt:-GIprime(SP(i)), [$1..1000]); # Robert Israel, May 20 2024

PARI
```
\\ See Links section.
```

A333825 Lexicographically earliest sequence of distinct positive integers, which when mapped onto a square spiral, gives a set without three distinct aligned points.

Original entry on oeis.org

1, 2, 3, 4, 17, 20, 22, 27, 45, 48, 67, 79, 80, 131, 135, 174, 180, 194, 201, 209, 236, 254, 312, 319, 394, 523, 644, 656, 706, 711, 733, 765, 766, 845, 848, 921, 922, 935, 1034, 1051, 1219, 1292, 1310, 1330, 1399, 1410, 1546, 1589, 1674, 1792, 1816, 1863

Offset: 1

Rémy Sigrist, Apr 07 2020

nonn

This sequence has similarities with A236266.

			The first terms, mapped onto a square spiral, are:
         *---*---*---*---*---*---*---*---*
         |                               |
         *   *---*---*---*---*---*---*   *
         |   |                       |   |
        67   *  17---*---*---*---*   *   *
         |   |   |               |   |   |
         *   *   *   *---4---3   *   *   *
         |   |   |   |       |   |   |   |
         *   *   *   *   1---2   *   *   *
         |   |   |   |           |   |   |
         *   *  20   *---*---*---*  27   *
         |   |   |                   |   |
         *   *   *--22---*---*---*---*   *
         |   |                           |
         *   *---*--45---*---*--48---*---*
         |
         *---*---*---*---*---*--79--80---*

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of (A174344(a(n)), A274923(a(n))) in the region -20000 <= x, y <= 20000
Rémy Sigrist, C# program for A333825

See A333835 and A333866 for a similar sequences.

Cf. A174344, A236266, A274923.

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 11, 21, 40, 47, 93, 180, 203, 397, 796, 1576, 1675, 3305, 6636, 13192, 14004, 27607, 55029, 110192, 220024, 226740, 450123, 898661, 1798700, 3594248, 3704800, 7354303, 14681369, 29349536, 58710640, 117394896, 119196748, 237492079

Offset: 0

Alec Jones and Peter Kagey, May 09 2020

nonn

The spiral track being used here is the same as in A274640, except that the starting cell here is indexed 0 (as in A274641).

The central cell gets index 0 (and we fill it in with the value a(0)=1).

			Spiral begins:
     3----2----1
     |         |
     5    1----1   47
     |              |
     8---11---21---40
a(11) = 47 = 1 + 1 + 5 + 40, the sum of the cells in its row and column.

Peter Kagey, Table of n, a(n) for n = 0..2499

Cf. A280027.
Cf. A180714, A214526, A274640, A278180, A304586, A307834, A334742.
x- and y-coordinates are given by A174344 and A274923, respectively.

\\ here P(n) returns A174344 and A274923 as pair.
P(n)={my(m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if(n<0, if(n<-m, [k, 3*k+n], [-k-n, k]), if(nAndrew Howroyd, May 09 2020

A340171 List of X-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340172 gives Y-coordinates.

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -2, -2, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -6, -6, -6, -6, -6, -6, -6, -6, -6, -6, -6, -5

Offset: 0

Rémy Sigrist, Dec 30 2020

sign

The odd function f such that f(n) = (a(n), A340172(n)) for any n >= 0 will visit exactly once every lattice point (so it is a bijection from Z to Z^2).

			The spiral starts as follows:
      +-----+-----+-----+-----+-----+
      .                             |
      .                             |
      .     +-----+-----+-----+     +
      .     |5     4     3    |2    |
      .     |                 |     |
            +     +-----+-----+     +
            |6    |      0     1    |     .
            |     |                 |     .
            +     +-----+-----+-----+     .
            |7                            .
            |                             .
            +-----+-----+-----+-----+-----+
             8     9     10    11    12    13
- so a(0) = a(3) = a(10) = 0,
-    a(1) = a(2) = a(11) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A340171
Index entries for sequences related to coordinates of 2D curves

Cf. A000217, A014105, A046092, A001105, A139274, A139275, A174344, A340172.

PARI
```
See Links section.
```

abs(a(n+1)-a(n)) + abs(A340172(n+1)-A340172(n)) = 1.
a(n) = A340172(n) iff n belongs to A001105.
a(n) = -A340172(n) iff n belongs to A046092.
a(n) = 2*A340172(n) iff n belongs to A139274.
2*a(n) = A340172(n) iff n belongs to A139275.
a(n) * A340172(n) = 0 iff n belongs to A000217.
a(n) = 0 iff n belongs to A014105.

A341278 The smallest spiral number not covered by any square in the minimal-sum square spiral tiling by n X n squares in A341363.

Original entry on oeis.org

67, 173, 25, 30, 42, 56, 72, 90, 110, 132, 156, 182, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 810, 860, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2067, 2159, 2253, 2349, 2447, 2547, 2649, 2753, 2859, 2967, 3077, 3189

Offset: 2

Scott R. Shannon, Feb 08 2021

nonn

The tilings with n=2 and n=3 are the only ones where the smallest uncovered square is not adjacent to the first centrally placed tile. The sequence starts at n=2 as a 1 X 1 square tiling leaves no squares uncovered.

See A341363 for other images with higher numbers of placed tiles.

Scott R. Shannon, Image of the tiling for n=2. The smallest uncovered square is 67. In this and other images the colors are graduated around the spectrum to show the squares relative placement order.
Scott R. Shannon, Image of the tiling for n=3. The smallest uncovered square is 173.
Scott R. Shannon, Image of the tiling for n=4. The smallest uncovered square is 25.
Scott R. Shannon, Image of the tiling for n=5. The smallest uncovered square is 30.
Scott R. Shannon, Image of the tiling for n=6. The smallest uncovered square is 42.

Cf. A341363, A341160, A341327, A340974, A174344, A274923, A296030, A275161.

A341363 Table read by antidiagonals: T(n, k) is the sum of the numbers inside the k-th square of size n X n when the square spiral is tiled with these squares, where each tile contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one tile.

Original entry on oeis.org

1, 2, 10, 3, 48, 45, 4, 60, 276, 136, 5, 68, 321, 928, 325, 6, 80, 368, 1040, 2349, 666, 7, 92, 384, 1168, 2575, 4984, 1225, 8, 100, 429, 1296, 2825, 5382, 9391, 2080, 9, 124, 456, 1388, 3075, 5816, 10030, 16228, 3321, 10, 128, 554, 1656, 3627, 6250, 10718, 17190, 26257, 5050

Offset: 1

Scott R. Shannon, Feb 10 2021

The terms for a given n tend to have larger jumps in value at one more than the square of the odd numbers, i.e., at k = (2*t+1)^2 + 1, t >= 0, due to the previous square filling a grid of squares containing (2*t+1)^2 squares. This forces the next square to move further away from the origin and into spiral arms containing larger numbers.

See A341278 for the smallest spiral number not covered by any square in each n X n tiling.

			The table begins:
     1,     2,     3,     4,     5,     6,     7,     8,     9,     10, ...
    10,    48,    60,    68,    80,    92,   100,   124,   128,    156, ...
    45,   276,   321,   368,   384,   429,   456,   554,   702,    803, ...
   136,   928,  1040,  1168,  1296,  1388,  1656,  1696,  1858,   2876, ...
   325,  2349,  2575,  2825,  3075,  3627,  3935,  4243,  4415,   7740, ...
   666,  4984,  5382,  5816,  6250,  8456,  9188,  9576, 10154,  14204, ...
  1225,  9391, 10030, 10718, 11406, 15006, 16260, 16737, 17627,  27701, ...
  2080, 16228, 17190, 18216, 19242, 24856, 26856, 27392, 28692,  49240, ...
  3321, 26257, 27636, 29096, 30556, 38998, 42010, 42561, 44383,  81527, ...
  5050, 40344, 42246, 44248, 46250, 58560, 62892, 63400, 65870, 127660, ...
  7381, 59459, 62002, 64666, 67330, 84806, 90808, 91201, 94459, 191129, ...
  ...
.
a(2,1) = 10 as the first square of size 2 X 2 is placed such that it covers the numbers 1,2,3,4, which sum to 10. This is the minimum possible sum.
a(2,2) = 48 as the second square of size 2 X 2 is placed such that it covers the numbers 5,6,18,19, which sum to 48. This is the minimum possible sum for such a square which does not use the previously covered numbers 1,2,3,4.
a(2,3) = 60 as the third square of size 2 X 2 is placed such that it covers the numbers 7,8,22,23, which sum to 60. This is the minimum possible sum for such a square which does not use the previously covered numbers 1,2,3,4,5,6,18,19.

Scott R. Shannon, Image for n=2, k = 1..2000. The image can be zoomed in to see the numbers of the square spiral. In this and other images the colors are graduated around the spectrum to show the squares relative placement order.
Scott R. Shannon, Image for n=4, k = 1..1000.
Scott R. Shannon, Image for n=5, k = 1..1000.
Scott R. Shannon, Image for n=7, k = 1..1000.
Scott R. Shannon, Image for n=10, k = 1..500.

Cf. A341160, A341327, A340974, A174344, A274923, A296030, A275161.

T(1,k) = k.

T(n,1) = n^2*(n^2+1)/2 = A000217(n^2).

A361486 Lexicographically earliest sequence of positive numbers on a square spiral such that no three equal numbers are collinear.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 1, 3, 3, 1, 4, 1, 4, 3, 5, 5, 1, 4, 3, 4, 5, 4, 4, 5, 6, 6, 7, 4, 4, 5, 5, 6, 2, 4, 1, 4, 5, 1, 6, 2, 6, 4, 6, 5, 5, 7, 2, 3, 4, 6, 5, 5, 7, 2, 3, 8, 1, 4, 3, 6, 7, 5, 5, 3, 5, 7, 6, 3, 1, 1, 7, 8, 7, 7, 4, 5, 8, 5, 9, 6, 6, 8, 7, 7, 6, 8, 9, 9, 3

Offset: 1

Scott R. Shannon, Mar 13 2023

The first term a(1) = 1 lies at the (0,0) origin while all other terms lie on integer coordinates.

			a(5) = 2 as a(3) = 1 and a(4) = 1 lie on the horizontal line y = 1 relative to the starting square (assuming a counter-clockwise spiral) so a(5) cannot be 1.
a(7) = 3 as a(5) = 2 and a(6) = 2 lie on the vertical line x = -1 so a(7) cannot be 2, while a(1) = 1 and a(3) = 1 lie on the line y = x so a(7) cannot be 1.
a(21) = 4 as a(18) = 3 and a(19) = 3 lie on the line x = -2, a(6) = 2 and a(15) = 2 lie on the line y = 2*x + 2, while a(1) = 1 and a(3) = 1 lie on the line y = x, so a(21) cannot be 1, 2 or 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 50000 terms on the square spiral. The values are scaled across the spectrum from red to violet to show their relative size. Zoom in to see the numbers.
Scott R. Shannon, Image highlighting the 1 valued terms in the first 50000 terms on the square spiral. Zoom in to see the numbers.

Cf. A274640, A229037, A174344, A274923, A346294.

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.

A368127 a(n) is the x-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the symmetrized variant of the distance-limited strip bijection described in A368126.

Original entry on oeis.org

0, 1, 0, -1, -1, -1, 0, 1, 1, 2, 2, 1, 0, 0, -1, -2, -3, -2, -2, -1, 0, 0, 1, 2, 3, 3, 3, 2, 2, 1, 0, -1, -2, -2, -3, -3, -4, -3, -3, -2, -2, -1, 0, 1, 2, 2, 3, 3, 4, 5, 4, 4, 3, 2, 1, 0, 0, -1, -1, -2, -3, -4, -4, -5, -6, -5, -4, -4, -3, -2, -1, 0, 0, 1, 1, 2, 3, 4, 4, 5, 6, 6, 6, 5, 4, 3

Offset: 0

Hugo Pfoertner, Jan 07 2024

sign

Hugo Pfoertner, Table of n, a(n) for n = 0..3000
Hugo Pfoertner, Plot of mapped spiral, using Plot 2.
Index entries for sequences related to coordinates of 2D curves

A368128 gives the corresponding y-coordinates.
Cf. A367150, A368126.
Analogous sequences, but without symmetrization: A367895, A367896.

\\ ax(n), ay(n) after Kevin Ryde's functions in A174344 and A274923.
\\ It is assumed that the PARI programs from A367150 and A368126 have been loaded and the functions defined there are available.
ax(n) = {my (m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if (n<0, if (n<-m, k, -k-n), if (n

cp's OEIS Frontend

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

A308412 Indices of Gaussian primes on a square spiral.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A333825 Lexicographically earliest sequence of distinct positive integers, which when mapped onto a square spiral, gives a set without three distinct aligned points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A334741 Fill an infinite square array by following a spiral around the origin; in the central cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that are in the same row or column as that cell.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A340171 List of X-coordinates of point moving along one of the arms of a counterclockwise double square spiral; A340172 gives Y-coordinates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A341278 The smallest spiral number not covered by any square in the minimal-sum square spiral tiling by n X n squares in A341363.

Views

Author

Keywords

Comments

Links

Crossrefs

A341363 Table read by antidiagonals: T(n, k) is the sum of the numbers inside the k-th square of size n X n when the square spiral is tiled with these squares, where each tile contains numbers which sum to the minimum possible value, and each number on the spiral can only be in one tile.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A361486 Lexicographically earliest sequence of positive numbers on a square spiral such that no three equal numbers are collinear.

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