A329231 The maximum number of times one reaches a single position during the grasshopper procedure.

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

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 Mathematics 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

Mathematics Stack Exchange User Vepir, Grasshopper jumping on circles

Crossrefs

Cf. A282442, A329230, A329232, A329233.

Sequence in context: A097794 A275494 A137683 * A259341 A369589 A350733

Adjacent sequences: A329228 A329229 A329230 * A329232 A329233 A329234

Keyword

nonn,walk

Author

Peter Kagey, Nov 08 2019

Status

approved