A329233 The time of the first counterclockwise step during the grasshopper procedure, or 0 if no counterclockwise steps occur.

0, 0, 2, 0, 4, 5, 4, 0, 4, 4, 6, 5, 8, 5, 5, 0, 13, 6, 10, 6, 6, 7, 12, 17, 7, 8, 7, 7, 16, 8, 16, 0, 8, 12, 8, 8, 20, 13, 9, 10, 25, 9, 22, 9, 9, 13, 24, 17, 10, 12, 11, 10, 28, 10, 10, 11, 12, 16, 30, 11, 32, 17, 11, 0, 11, 11, 34, 12, 14, 13, 36, 12, 41

Offset: 1

Author

Peter Kagey, Nov 10 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.)

Either a(n) = 0 or a(n) >= A003056(n).

Conjecture: a(3^n) = A087503(n-1) + 1 for n > 0. (Checked up to 3^12.) - Peter Kagey, Nov 28 2019

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Mathematics Stack Exchange User Vepir, Grasshopper jumping on circles

Crossrefs

Cf. A003056, A329230, A329231, A329232.