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A329231 The maximum number of times one reaches a single position during the grasshopper procedure. 4
1, 1, 1, 1, 3, 2, 1, 1, 4, 3, 3, 3, 3, 3, 3, 1, 5, 2, 3, 3, 4, 3, 5, 3, 3, 4, 5, 3, 4, 4, 4, 1, 4, 4, 3, 4, 4, 3, 3, 5, 4, 5, 3, 3, 4, 4, 5, 4, 6, 4, 5, 4, 5, 4, 5, 4, 4, 4, 5, 5, 4, 5, 5, 1, 4, 4, 5, 3, 4, 5, 5, 4, 4, 7, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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 Math Stack Exchange link for more details.)

a(n) >= (A329230(n)-1)/(n-1).

Least values of n such that a(n) = 1, 2, 3, etc are 1, 6, 5, 9, 17, 49, 74, 198, 688, 1745 etc.

Conjecture: a(n) = 1 if and only if n = 3, n = 7, or n = 2^k for some k.

Conjecture: The largest values of n for which a(n) = 2, 3, 4, 5 respectively are n = 18, 68, 381, 1972.

If the second conjecture is true, then 2, 3, 4, and 5 appear 2, 19, 87, and 313 times respectively.

Conjecture: Every integer greater than 1 appears in this sequence a finite number of times.

LINKS

Peter Kagey, Table of n, a(n) for n = 1..2048

Math Stack Exchange User Vepir, Grasshopper jumping on circles

CROSSREFS

Cf. A282442, A329230, A329232, A329233.

Sequence in context: A097794 A275494 A137683 * A259341 A046225 A269233

Adjacent sequences:  A329228 A329229 A329230 * A329232 A329233 A329234

KEYWORD

nonn,walk

AUTHOR

Peter Kagey, Nov 08 2019

STATUS

approved

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Last modified September 18 04:44 EDT 2020. Contains 337165 sequences. (Running on oeis4.)