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A351042
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Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n.
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8
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9, 12, 13, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 34, 34, 36, 36, 37
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OFFSET
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1,1
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COMMENTS
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The car moves according to the rules of the game of Racetrack with von Neumann neighborhood, i.e., if P, Q, and R are three successive positions of the car, one coordinate of the second difference (acceleration vector) P - 2Q + R must be 0, and the other 1, 0, or -1. The car starts with zero velocity at a point (x,0) for some integer x >= n, and finishes when it passes, or lands on, the positive x-axis after a complete counterclockwise lap around the origin. The line segments between successive positions must be outside or on the circle with center in (0,0) and radius n.
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LINKS
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FORMULA
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EXAMPLE
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The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.
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a(1) = 9:
. 3 2 . .
4 . . 1 .
5 . * 0 9
. 6 7 8 .
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a(2) = 12:
. 4 3 2 . .
5 . . . 1 .
6 . * . 0 12
7 . . . 11 .
. 8 9 10 . .
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a(3) = 13:
. . . 4 . 3 . . . .
. 5 . . . . . 2 . .
6 . . . . . . . 1 .
7 . . . * . . . 0 13
8 . . . . . . . . .
. 9 . . . . . 12 . .
. . . 10 . 11 . . . .
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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