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A329230
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The number of steps it takes to reach all n positions around a circle during the grasshopper procedure.
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4
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1, 2, 3, 4, 10, 9, 7, 8, 23, 17, 16, 21, 20, 27, 29, 16, 58, 25, 28, 30, 40, 43, 88, 50, 40, 55, 87, 50, 62, 59, 70, 32, 81, 72, 67, 91, 84, 73, 71, 125, 107, 113, 69, 88, 148, 116, 135, 113, 158, 95, 137, 114, 182, 123, 174, 166, 112, 146, 215, 173, 126, 171
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OFFSET
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1,2
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COMMENTS
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The grasshopper procedure: n positions are evenly spaced around a circle, a grasshopper hops randomly to any position, after the k-th hop, the grasshopper looks clockwise and counterclockwise k positions. If one of the positions has been visited less often then the other, it hops there; if both positions have been visited an equal number of times, it hops k steps in the clockwise position. (See Mathematics Stack Exchange link for more details.)
In the Mathematics Stack Exchange link, the author conjectures that a(n) = n if and only if n = 3, n = 7, or n = 2^k for some nonnegative k.
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LINKS
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EXAMPLE
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For n = 5 the a(5) = 10 steps are:
[0,0,0,0,0] (randomly step to first position)
[1,0,0,0,0] (length 1 clockwise (right) step)
[1,1,0,0,0] (length 2 clockwise (right) step)
[1,1,0,1,0] (length 3 clockwise (right) step)
[1,2,0,1,0] (length 4 counterclockwise (left) step)
[1,2,1,1,0] (length 5 clockwise (right) step)
[1,2,2,1,0] (length 6 clockwise (right) step)
[1,2,2,2,0] (length 7 clockwise (right) step)
[2,2,2,2,0] (length 8 clockwise (right) step)
[2,2,2,3,0] (length 9 counterclockwise (left) step)
[2,2,2,3,1]
For example, the length 4 counterclockwise step occurs because stepping clockwise would result in landing in a position which has been visited once, and stepping counterclockwise would result in landing in a position which has not been visited before.
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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