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).

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

Author

Hugo Pfoertner, Apr 25 2019

Keywords

• nonn

Comments

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.

Examples

			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
		

Links

• 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.

Crossrefs

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

Programs

PARI

\\ See Pfoertner link.