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

0,2

Comments

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... .

Links

Hugo Pfoertner, Table of n, a(n) for n = 0..10001

Hugo Pfoertner, PARI program.

Rainer Rosenthal and Hugo Pfoertner, A367150 compared to A307110.

Example

   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.

Prog

(PARI) \\ See Pfoertner link.

Crossrefs

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

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

Sequence in context: A328941 A205691 A002141 * A368121 A368126 A361472

Adjacent sequences: A367147 A367148 A367149 * A367151 A367152 A367153

Keyword

nonn,changed

Author

Rainer Rosenthal and Hugo Pfoertner, Nov 22 2023

Status

approved