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

Author

Peter Kagey, Nov 08 2019

Keywords

• nonn
• walk

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.