A307110 -id:A307110 - cp's OEIS frontend

A367150 Results of the strip bijection as described in A307110 with subsequent reassignment of the pair connections at all locations, in which 4 points of a unit square in one grid are mapped to a unit square in the other (rotated by Pi/4) grid in such a way that the maximum distance of the two points in the 4 assigned pairs is minimized.

0, 5, 6, 7, 8, 2, 3, 4, 1, 13, 15, 17, 19, 14, 10, 16, 11, 18, 12, 20, 9, 26, 27, 28, 25, 21, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 37, 30, 31, 32, 33, 34, 35, 36, 29, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 61, 46, 47, 48, 45, 50, 51, 52, 53, 54, 55

Offset: 0

Rainer Rosenthal and Hugo Pfoertner, Nov 22 2023

nonn

The strip bijection of A307110 assigns each grid point in one grid to a unique grid point in the rotated grid. The mapping therefore corresponds to a permutation of the nonnegative integers. Approximately 2/3 of the grid points are mapped in such a way that 4 points that form a unit square in the original grid also form a unit square after being mapped onto the rotated grid. We call this a stable (grid) cell under the bijection map. The method differs from that used in A307731 in that for each stable cell it is tried whether the maximum of the 4 pair distances resulting from the application of strip bijection can be reduced by a cyclic rotation of the connections. The one of the two assignments by cyclic connection change is selected that provides a smaller maximum of the 4 distances in the pairs assigned to each other. In contrast, a cyclic rotation of the connections is only carried out in the method of A307731 if the maximum of the 4 distances exceeds the upper limit of the bijection distance of sqrt(5)*sin(Pi/8)=0.855706... .

			   n   i = A305575(n)
   |   |   j = A305576(n)
   |   |   |   A307110(n)
   |   |   |   |  k   m  distance_A307110
   |   |   |   |  |   |    |      a(n)  k'  m' distance after
   |   |   |   |  |   |    |        |   |   |  reconnecting
   0   0   0   0  0   0  0.0000     0   0   0   0.0000
   1   1   0   1  1   0  0.7654 L   5   1   1   0.4142  r
   2   0   1   6 -1   1  0.4142     6  -1   1   0.4142
   3  -1   0   3 -1   0  0.7654 L   7  -1  -1   0.4142  r
   4   0  -1   8  1  -1  0.4142     8   1  -1   0.4142
   5   1   1   2  0   1  0.4142     2   0   1   0.4142
   6  -1   1  11 -2   0  0.5858     3  -1   0   0.4142  r
   7  -1  -1   4  0  -1  0.4142     4   0  -1   0.4142
   8   1  -1   9  2   0  0.5858     1   1   0   0.4142  r
   9   2   0   5  1   1  0.5858    13   2   1   0.7174  r
  10   0   2  15 -1   2  0.7174    15  -1   2   0.7174
  11  -2   0   7 -1  -1  0.5858    17  -2  -1   0.7174  r
  13   2   1                improved by reconnecting
  15  -1   2         L = 0.7654      ->         0.7174
  17  -2  -1
See the linked file for a visualization of the differences from A307110.

Hugo Pfoertner, Table of n, a(n) for n = 0..10001
Hugo Pfoertner, PARI program.
Rainer Rosenthal and Hugo Pfoertner, A367150 compared to A307110.

Cf. A305575, A305576 (enumeration of the grid points in the square lattice).

Cf. A307110, A307731, A367146, A367895, A367896.

PARI
```
\\ See Pfoertner link.
```

A307731 Results of strip bijection as described in A307110 with additional application of local repairs to reduce the maximum wobbling distance S from S1=cos(Pi/8) to S2=sqrt(5)*sin(Pi/8).

Original entry on oeis.org

0, 1, 6, 3, 8, 2, 11, 4, 9, 5, 15, 7, 19, 14, 10, 16, 17, 18, 12, 20, 13, 26, 27, 28, 25, 21, 22, 23, 24, 38, 31, 40, 33, 42, 35, 44, 29, 30, 51, 32, 53, 34, 55, 36, 49, 57, 58, 59, 60, 62, 39, 64, 41, 66, 43, 68, 37, 46, 47, 48, 45, 50, 63, 52, 65, 54, 67

Offset: 0

Hugo Pfoertner, Apr 25 2019

nonn

The terms visible in the data section are identical with those of A307110. The first difference occurs at a(124)=141, A307110(124)=125.
The wobbling distance S is the mutual Euclidean distance of the pairs matched by a bijection.
.
  - - - G - -\- - - - - - / - G - - - - -\- - - - - G -/- - - - - - - - - G
        |  +   \       /      |             \      +|/                    |
        |      + \   /        |               \   + |                     |
        |          H          |                 H   |                     |
        |       /    \        |               /   \ |                     |
        |     /        \      .     .   .   /       |\                    |
        |  /           . \    |           /  .      |  \                 /|
        |/                 \  |        /        .   |     \            /  |
       /|           .        \|      /            . |        \       /    |
    /   |                     |\   /               .|          \   /      |
  H +   |        .            | +H # # # #          .            H + +    |
  - \ + G - - - - - - - - - - G+ -\- - - - # # # # #G.- - - - - - -\- -+ +G
      \ |       .            #| \   \              +| .        /     \    |
        \                   # |  D    \            +|        /          \ |
        | \    .          #   |   \     \          +|      /              \
        |   \   <--------r=S1------C      \       + |  ./                 |
        |     \       #       |     B        \    + | /                   |
        |       .   #         |      B          \ + /                     |
        |        .H+          |       B           H |.                    |
        |       /   \++       |        B        /  #.                     |
        |     /   .   \++     |        B      /    #| \                   |
        |   /       .   \+++  |         B   /     . #   \                 |
  - - - G / - - - - - . - \ -+G - - - - -B/ - - .  +G - - \ - - - - - - - G
      + /               .   \ |#        / B    +E+. | .     \           + /
     +/ |                     . #     /.  +M+       |       . \      + /  |
   +/   |                     | \#  / +E+   b       |          .\   +/    |
  H     |                     |   H+ .       b      |            .H       |
    \   |                     | /  .\         b     |           /  .\     |
      \ |                     /   .   \        b    |         /     . \   |
        \                   / |  .       \      b   |      /            \ |
        | \               /   | .          \     b  |   /             .   \
        |   \           /     |.             \    b | /                   |
        |     \       /       |.                \  c--------r=S1------>.  |
  - - - G+++++- \ - / - - - - G.- - - - - - - - - HdG - - - - - - - - -.- G
        |     ++++H           +.                /   \                  .  |
        |       /   \         |+              /     | \                   |
        |    /         \      |+.           /       |   \             .   |
        |  /             \    | +         /         |     \          .    /
        |/                  \ | + .     /           |       \       .   / |
       /                      \  + .  /             |          \   .  /   |
    /   |                     | \ + .               |             . /     |
  H     |                     |   H   .             |           . H       |
    \   |                     | /   \       .       |       .   /    +    |
      \ |                     /        \            .         /         + |
  - - - G - - - - - - - - - / G - - - - -\- - - - - G - - - / - - - - - - G
.
The ASCII graphics above shows the situation after the application of the strip bijection, as it is described in A307110, for a position in the grids containing a "long" junction exceeding the length S2. The linked graphics file "Construction of repair" shows a similar configuration, but without labels.
All junctions resulting from the strip bijection are marked by plus signs. The long junction is marked by embedded letters "E". There are 6 possible orientations of E-junctions (called E for short), but the method for their elimination is identical for all cases.
The target of the method is to achieve a local reconnecting, which replaces 4 junctions by circularly shifted new junctions. To determine the affected grid points, the following steps are performed:
From the midpoint (marked by M in the figure) of E construct a bisecting line Bb perpendicular to E. Draw two circles, one on each side of E with centers on B and b at distance S1 from M. E is a tangent at M of these circles with radius r = S1. The two circles are marked by dots ".." in the figure.
For the two circle centers C and c determine the distances D and d of the respective closest grid points in lattice G. The position (c) of the circle center, for which this minimum distance is smaller, indicates on which side of E no reconnecting is required. A circle with radius S1 around c contains only one grid point of G and one of H. All other grid points of both lattices lie outside of this circle.
The side of E with the larger distance between circle center and closest grid point is where the circular shift of junctions is to be performed. The circle around C with radius r = S1 contains 3 grid points of lattice G and 3 grid points of lattice H.
After having found c, it is possible to replace the geometric determination of the 3 grid point pairs on the opposite side of E by a lookup in a table of differences between the coordinates of M and c rounded to nearest integers, leading to a unique identification of the 6 occurring cases. The function "repair" in the PARI program implements this selection.
The 4 new junctions are marked by "###" in the figure. They replace the 4 previous "+++" junctions, including the long junction E. The maximum of their lengths does not exceed S2, approaching S2 for length of E approaching S1. The limiting case for the 4 rearranged junctions are two of length S2 and two of length sin(Pi/8) = 0.38268...
The described repair is applied to all occurrences of bijection distances exceeding S2 within the overlay of the two lattices. Numerical experiments with random points on square lattices of huge size show that approximately 0.956 % (roughly 1/105) of the grid points lead to a bijection distance S > S2 after the application of the strip bijection. No counterexample for the validity of the method is known, but a formal proof is missing.
In the ring-wise one-dimensional mapping of the bijection as given by A307110, the first affected position is n = 124. The table in the example section shows the corresponding changes for this earliest repair together with the listing of another repair with different orientation of E.
All affected index positions have to be exchanged in the one-dimensional list. Due to the occurrence frequency of E-junctions the current sequence is expected to differ from A307110 for roughly 4% of the terms.
The PARI program provided as external file is self-contained, including the code for generation of the rings used for 1d-mapping, A305575 and A305576, and the code for the strip bijection of A307110. To generate a b-file of 10000 terms, the corresponding code lines at the end of the program have to be activated.

			The table shows the first re-matched pairs of grid points together with the result of the unmodified strip bijection:
    Grid G          Grid H             Grid H rotated
   n     i    j    a(n)   k    m   (k,m) rotated by -Pi/4  distance of
                                                           matched points
  124   -6    2    141   -6   -3    -6.363961   2.121320   0.383648
  140   -6    3    125   -6   -2    -5.656854   2.828427   0.383649
  180   -7    3    173   -7   -2    -6.363961   3.535534   0.831470 < S2
  172   -7    2    181   -7   -3    -7.071068   2.828427   0.831470 < S2
  ...
  266   -6    7    256   -9    1    -5.656854   7.071068   0.350428
  309   -6    8    320  -10    1    -6.363961   7.778174   0.426232
  279   -5    8    328  -10    2    -5.656854   8.485281   0.816673 < S2
  235   -5    7    268   -9    2    -4.949747   7.778174   0.779795 < S2
compared to (unmodified):                                        S2=0.855706..
                A307110(n)
  124   -6    2    125   -6   -2    -5.656854   2.828427   0.896683 > S2
  140   -6    3    173   -7   -2    -6.363961   3.535534   0.647506
  180   -7    3    181   -7   -3    -7.071068   2.828427   0.185709
  172   -7    2    141   -6   -3    -6.363961   2.121320   0.647506
  ...
  266   -6    7    320  -10    1    -6.363961   7.778174   0.859083 > S2
  309   -6    8    328  -10    2    -5.656854   8.485281   0.594346
  279   -5    8    268   -9    2    -4.949747   7.778174   0.227446
  235   -5    7    256   -9    1    -5.656854   7.071068   0.660688

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Hugo Pfoertner, Construction of repair.
Hugo Pfoertner, Repair of strip bijection.
Hugo Pfoertner, Illustration of repair locations, 10000 grid points. Zoom in to see details.
Hugo Pfoertner, PARI program for A307731.

Cf. A144981, A305575, A305576, A307110, A367150, A386241.

PARI
```
\\ See Pfoertner link.
```

A362955 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 distance-limited strip bijection described in A307110.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, May 10 2023

Hugo Pfoertner, Table of n, a(n) for n = 0..3000
Hugo Pfoertner, Plot of mapped spiral, using Plot 2.

A362956 gives the corresponding y-coordinates.

Cf. A174344, A274923, A307110, A307731, A367150, A367895, A367896.

\\ ax(n), ay(n) after Kevin Ryde's functions in A174344 and A274923,
\\ p(i,j) given in A307110
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

A362956 a(n) is the y-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the distance-limited strip bijection described in A307110.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, May 10 2023

sign

Hugo Pfoertner, Table of n, a(n) for n = 0..3000
Hugo Pfoertner, Plot of mapped spiral, using Plot 2.

A362955 gives the corresponding x-coordinates.

Cf. A174344, A274923, A307110, A307731.

\\ for functions ax, ay, p see A362955
for (k=0, 81, print1 (p(ax(k),ay(k))[2]", "))

A368121 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between 0 and Pi/4 of the pair of rotated grids created by the strip bijection of A307110.

Original entry on oeis.org

Offset: 0

Hugo Pfoertner, Jan 05 2024

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Klaus Nagel, Symmetrized strip bijection, with mirrored repetition of sector 0.
Hugo Pfoertner, Distance minimization applied to symmetrized strip bijection, using sector between 0 and Pi/4.
Hugo Pfoertner, PARI program.

A305575, A305576 are used for enumeration of the grid points.

Cf. A307110, A367150, A368124, A368125, A368126.

PARI
```
\\ See Pfoertner link.
```

A368126 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between Pi/4 and Pi/2 of the pair of rotated grids created by the strip bijection of A307110.

Original entry on oeis.org

Offset: 0

Hugo Pfoertner, Jan 05 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Klaus Nagel, Symmetrized strip bijection, with mirrored repetition of sector 1.
Hugo Pfoertner, Distance minimization applied to symmetrized strip bijection, using sector between Pi/4 and Pi/2.
Hugo Pfoertner, PARI program.

A305575, A305576 are used to enumerate the grid points in rings of increasing radius.

Cf. A307110, A367150, A368121, A368129, A368130.

PARI
```
\\ See Pfoertner link.
```

A363760 Cycle lengths obtained by repeated application of Klaus Nagel's strip bijection, as described in A307110.

Original entry on oeis.org

1, 8, 9, 10, 40, 72, 106, 218, 256, 408, 424, 872, 1178, 2336, 2522, 2952, 4712, 10088, 13290, 26648, 28906, 33784, 53160, 115624, 150842, 303784, 330138, 385624, 603368, 1320552, 1716170, 3462216, 3765322, 4397144, 6864680, 15061288, 19543834, 39454792, 42921274, 50118936, 78175336, 171685096

Offset: 1

Hugo Pfoertner, Jun 26 2023

nonn

Description provided by Klaus Nagel: (Start)
The strip bijection s (A307110) maps a point P[i,j] from a Z X Z grid to Q[u,v] taken from a second grid obtained from the first one by a rotation by Pi/4 around the origin. The coordinates [i,j] and [u,v] refer to the respective grids. If [u,v] also are considered as coordinates of the first grid, the mapping [i,j] --> [u,v] establishes a permutation of the grid points of Z X Z.
Cycles of this permutation are evaluated; the sequence shows the occurring cycle lengths L.
Q[u,v] is located close to P[i,j]. Changing the reference to the other grid causes a rotation by Pi/4. Hence after eight permutation steps any point should return to the vicinity of its starting position. (End)
Therefore the provided visualizations also include graphs showing only every 8th point for cycles with L divisible by 8.
Examples of cycles with lengths > 10^9 are L = 2536863994 for the starting position [1761546, 1379978], L = 5574310746 for start [5207814, 6746677], and L = 6508768664 for start [7983336, 8380845].

			a(1) = 1: p(0, 0) -> [0, 0], p(1, 0) -> [1, 0]. Points mapped onto themselves.
a(2) = 8: [0, 1] -> [-1, 1] -> [-2, 0] -> [-1, -1] -> [0, -1] -> [1, -1] -> [2, 0] -> [1, 1] ->  [0, 1].
a(3) = 9: [1, 6] -> [-3, 5] -> [-6, 2] -> [-6, -2] -> [-3, -5] -> [1, -6] -> [5, -4] -> [6, 0] -> [5, 4] -> [1, 6].
a(4) = 10: [0, 2] -> [-1, 2] -> [-2, 1] -> [-2, -1] -> [-1, -2] -> [0, -2] -> [1, -2] -> [2, -1] -> [2, 1] -> [1, 2] -> [0, 2].
List of start points and corresponding cycle lengths:
    y  0   1   2   3   4  5   6   7   8   9  10  11  12  13  14 15 16
   x \---------------------------------------------------------------
   0 | 1   8  10   8   8 40   8   8   8  40   8   8 106   8   8 40  8
   1 | 1   8  10   8   8 40   9  40   8   8 106  40 106   8   8 40  8
   2 | 8  10   8   8   8  8   8   8   8   8  40 106   8   8   8  8 40
   3 | 8   8   8   8  40  9   8   8   8   8   8   8   8 106   8  8  8
   4 | 8   8   8  40   8 40   8   8   8   8   8   8   8   8 106  8  8
   5 | 8  40   8  40   9  8   8   8   8   8   8   8   8   8 106  8  8
   6 | 9  40   9   8   8 40   8  40 106  40 106   8   8   8 106 72  8
   7 | 8   8   8   8   8  8   8   8  40 106   8 106   8 106   8  8 72
   8 |40   8   8   8   8  8  40 106   8 106   8   8   8   8   8  8  8
   9 | 8   8   8   8   8  8 106  40 106   8   8   8   8   8   8  8  8
  10 | 8  40 106   8   8  8   8   8   8   8   8  40   8  40   8  8 72
  11 |40 106  40   8   8  8   8 106   8   8  40   8   8   8  40 72  8
  12 | 8 106   8 106   8  8   8 106   8   8  40   8   8   8  40  8  8
  13 | 8   8   8 106   8  8   8 106   8   8  40   8   8   8  40  8  8
  14 | 8   8   8   8 106  8 106   8   8   8   8  40   8  40   8  8  8
  15 | 8  40   8   8   8  8   8  72   8   8  72   8   8   8   8  8 40
  16 | 8   8  40   8   8  8  72   8   8   8   8  72   8   8   8 40  8
.
a(9) = 256: See links to animated visualizations.

Hugo Pfoertner, Animated visualization of cycle with L=256, all visited points shown.
Hugo Pfoertner, Animated visualization of cycle with L=256, every 8th visited point shown.
Hugo Pfoertner, Visualization of all terms from L=40 to L=53160. Zoom in to see details.
Hugo Pfoertner, Illustration of a(24) = 115624.
Hugo Pfoertner, Illustration of a(25) = 150842.
Hugo Pfoertner, Illustration of a(26) = 303784.
Hugo Pfoertner, Illustration of a(27) = 330138.
Hugo Pfoertner, Illustration of a(28) = 385624.
Hugo Pfoertner, Illustration of a(29) = 603368.
Hugo Pfoertner, Illustrations of a cycle of length 6508768664, including zoom images of the self-similar path details, December 2023.
Hugo Pfoertner, Examples of starting points for all known cycle lengths, July 2023.

Cf. A362955, A362956, A367146, A367893.

Cf. A367148 (analog of this sequence, but for the triangular lattice).

C=cos(Pi/8); S=sin(Pi/8); T=S/C; \\ Global constants
\\ The mapping function p
\\ PARI's default precision of 38 digits is sufficient up to abs({x,y})<10^17
p(i,j) = {my (gx=i*C-j*S, gy=i*S+j*C,k, xm, ym, v=[0,0]); k=round(gy/C); ym=C*k;xm=gx+(gy-ym)*T; v[1]=round((xm-ym*T)*C); v[2]=round((ym+v[1]*S)/C); v};
\\ cycle length
cycle(v) = {my (n=1, w=p(v[1],v[2])); while (w!=v, n++; w=p(w[1],w[2])); n};
a363760 (rmax) = {my (L=List()); for (x=0, rmax, for(y=x, rmax, my(c=cycle([x,y])); if(setsearch(L,c)==0, listput(L,c); listsort(L,1)))); L};
a363760(500) \\ takes a few minutes, terms up to a(19), check completeness of list with larger rmax

A367893 Length of cycles obtained by repeated application of the strip bijection for the square lattice (A307110), sorted by increasing minimum radius visited by any cycle of this length.

Original entry on oeis.org

1, 8, 10, 40, 9, 106, 72, 408, 218, 256, 1178, 424, 872, 2336, 2522, 2952, 13290, 4712, 10088, 26648, 28906, 33784, 150842, 53160, 115624, 303784, 330138, 385624

Offset: 1

Hugo Pfoertner, Dec 11 2023

Hugo Pfoertner, Examples of points at minimum radius.
Hugo Pfoertner, Illustration of all cycles with minimum radius up to 1000.

A permutation of A363760.

Cf. A307110, A367146.

A367146 Cycle lengths obtained by repeated application of the distance-minimizing variant of the strip bijection for the square lattice described in A367150.

Original entry on oeis.org

1, 8, 12, 24, 25, 56, 120, 152, 154, 200, 217, 376, 464, 568, 616, 1242, 1368, 1624, 1736, 1945, 4376, 4968, 5176, 10682, 13016, 14152, 15560, 17497, 40376, 42728, 46648, 94234, 120664, 125320, 139976, 157465, 367544, 376936, 419896, 840570, 1100760, 1119496, 1259720

Offset: 1

Hugo Pfoertner, Nov 25 2023

nonn

See the description in the similar A363760 for more information.

			a(1) = 1: D(0,0) -> [0,0];
a(2) = 8: [1,0] -> [1,1] -> [0,1] -> [-1,1] -> [-1,0] -> [-1,-1] -> [0,-1] -> [1,-1] -> [1,0];
a(3) = 12: [2,0] -> [2,1] -> [1,2] -> [0,2] -> [-1,2] -> [-2,1] -> [-2,0] -> [-2,-1] -> [-1,-2] -> [0,-2] -> [1,-2] -> [2,-1] -> [2,0].
List of start points and corresponding cycle lengths:
    y   0  1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16
   x \------------------------------------------------------------------
   0 |  1  8  12   8   8   8   8   8   8  25   8   8   8   8   8  24   8
   1 |  8  8  12   8   8   8   8   8   8 154   8   8   8   8   8  24   8
   2 | 12 12   8   8   8   8   8  25  25 154 154   8   8   8   8   8  24
   3 |  8  8   8   8   8   8  25  25   8   8 154 154 154 154   8   8   8
   4 |  8  8   8   8   8   8   8  25   8   8 154   8   8   8 154   8   8
   5 |  8  8   8   8   8   8   8 154 154 154 154   8   8   8 154   8 152
   6 |  8  8   8   8   8   8   8  25   8   8 154   8   8   8 154 152   8
   7 |  8  8  25  25  25  25 154   8   8   8   8 154 154 154   8 152   8
   8 |  8  8  25   8   8 154   8   8   8   8   8   8   8   8   8 152   8
   9 |154 25 154   8   8 154 154   8   8   8   8   8   8   8   8 152   8
  10 |  8  8 154 154 154 154 154   8   8   8   8  24  24  24   8 152   8
  11 |  8  8   8 154   8   8   8 154   8   8  24   8   8   8  24 152   8
  12 |  8  8   8 154   8   8   8 154   8   8  24   8   8   8  24   8 152
  13 |  8  8   8 154   8   8   8 154   8   8  24   8   8   8  24   8   8
  14 |  8  8   8   8 154 154 154   8   8   8   8  24  24  24   8   8   8
  15 | 24 24   8   8   8   8 152 152 152 152 152 152   8   8   8   8  24
  16 |  8  8  24   8   8 152   8   8   8 152   8   8 152   8   8  24   8

Hugo Pfoertner, Examples of starting points for all known cycle lengths, December 2023.
Hugo Pfoertner, Visualization of some selected orbits with lengths from L=24 to L=918330056, December 2023.

Cf. A307110, A363760, A367150, A367894.

\\ It is assumed that the PARI program from A367150 has been loaded and the functions defined there are available.
cycle(v) = {my (n=1, w=BijectionD(v)); while (w!=v, n++; w=BijectionD(w)); n};
a367146(rmax=205) = {my (L=List()); for (x=0, rmax, for(y=x, rmax, my(c=cycle([x, y])); if(setsearch(L, c)==0, listput(L, c); listsort(L, 1)))); L};
a367146() \\ produces terms up to a(18)=1624 in about 5 minutes run time.

cp's OEIS Frontend

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A307731 Results of strip bijection as described in A307110 with additional application of local repairs to reduce the maximum wobbling distance S from S1=cos(Pi/8) to S2=sqrt(5)*sin(Pi/8).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A362955 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 distance-limited strip bijection described in A307110.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A362956 a(n) is the y-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the distance-limited strip bijection described in A307110.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A368121 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between 0 and Pi/4 of the pair of rotated grids created by the strip bijection of A307110.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A368126 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between Pi/4 and Pi/2 of the pair of rotated grids created by the strip bijection of A307110.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A363760 Cycle lengths obtained by repeated application of Klaus Nagel's strip bijection, as described in A307110.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A367893 Length of cycles obtained by repeated application of the strip bijection for the square lattice (A307110), sorted by increasing minimum radius visited by any cycle of this length.

Views

Author

Keywords

Links

Crossrefs

A367146 Cycle lengths obtained by repeated application of the distance-minimizing variant of the strip bijection for the square lattice described in A367150.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A367895 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 variant of the distance-limited strip bijection described in A367150.

Views