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A367150
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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.
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16
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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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... .
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LINKS
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EXAMPLE
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| | | | | | | 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.
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PROG
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(PARI) See link.
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CROSSREFS
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Cf. A305575, A305576 (enumeration of the grid points in the square lattice).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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