A367146 -id:A367146 - cp's OEIS frontend

A368124 A variant of A367146 with application of the distance minimization to the first of two symmetrized versions of the strip bijection between two square lattices as described in A368121.

1, 8, 12, 24, 60, 72, 168, 216, 264, 300, 624, 1560, 1692, 1752, 2232, 4824, 9804, 12456, 13080, 17064, 35040, 57084, 92184, 92952, 123096, 244584, 332652, 639192, 651432, 855240, 1660752

Offset: 1

Hugo Pfoertner, Jan 01 2024

Apparently, a(n) == 0 (mod 4) for n > 1. For cycles, whose lengths are multiples of 8, the visited points form 8 separated islands.

Larger terms are 4293336, 4462104, 5787768, 11050488, 28333080, 38414184, 72397248.

Hugo Pfoertner, Example of one of the 8 islands in orbit with L=11050488.
Hugo Pfoertner, Illustration of the orbits corresponding to the terms a(2)-a(24) of A368125, (2024).

Cf. A307110, A367146, A367150, A368121.
A368125 is a permutation of this sequence.
A368129 is the analog for the second symmetrized version of the strip bijection.

\\ Uses definitions and functions from
\\ a367150_PARI.txt and a368121_PARI.txt
cycle(v) = {my (n=1, w=BijectionD(v, BijectionK)); while (w!=v, n++; w=BijectionD(w,BijectionK)); n};
a368124(rmax=205) = {my (L=List()); for (r2=0, rmax^2, for (x=0, sqrtint(r2), my (y2=r2-x^2,y); if (issquare(y2,&y), if(x>=y, my (c=cycle([x,y])); if (setsearch(L,c)==0, print([c,[x,y],sqrt(x^2+y^2)],", "); listput(L,c); listsort(L,1)))))); L};
a368124() \\ Terms < 1000

A368129 A variant of A367146 with application of the distance minimization to the second of two symmetrized versions of the strip bijection between two square lattices as described in A368126.

Original entry on oeis.org

1, 8, 12, 24, 72, 156, 168, 216, 264, 624, 1560, 1752, 1836, 2232, 4824, 12456, 13080, 16380, 17064, 35040, 92184, 92952, 123096, 128844, 244584, 639192, 651432, 855240, 945756

Offset: 1

Hugo Pfoertner, Jan 03 2024

Apparently, a(n) == 0 (mod 4) for n > 1. For cycles, whose lengths are multiples of 8, the visited points form 8 separated islands.

Larger terms are 1660752, 4293336, 4462104, 5787768, 6647916, 11050488, 28333080, 38414184, 45366204, 184427544.

			See files linked in A368130 for visualization of orbits.

Hugo Pfoertner, Example of one of the 8 islands in orbit with L=184427544.

Cf. A307110, A367146, A367150, A368126.
A368130 is a permutation of this sequence.
A368124 is the analog for the first symmetrized version of the strip bijection.

\\ Uses definitions and functions from
\\ a367150_PARI.txt and a368126_PARI.txt
cycle(v) = {my (n=1, w=BijectionD(v, Bijectionk)); while (w!=v, n++; w=BijectionD(w, Bijectionk)); n};
a368129(rmax=235) = {my (L=List()); for (r2=0, rmax^2, for (x=0, sqrtint(r2), my (y2=r2-x^2, y); if (issquare(y2, &y), if(x>=y, my (c=cycle([x, y])); if (setsearch(L, c)==0, print([c, [x, y], sqrt(x^2+y^2)], ", "); listput(L, c); listsort(L, 1)))))); L};
a368129() \\ Terms < 1000, takes 5-10 minutes CPU time

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.

Original entry on oeis.org

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.
```

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

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A368124 A variant of A367146 with application of the distance minimization to the first of two symmetrized versions of the strip bijection between two square lattices as described in A368121.

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A368129 A variant of A367146 with application of the distance minimization to the second of two symmetrized versions of the strip bijection between two square lattices as described in A368126.

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

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A367894 Length of cycles obtained by repeated application of the distance-minimizing strip bijection for the square lattice (A367150), sorted by increasing minimum radius visited by any cycle of this length.

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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.

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