This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329231 #20 Aug 12 2022 09:23:51 %S A329231 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, %T A329231 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, %U A329231 4,5,5,4,4,7,4,4,4,5,5,5,4,4,4,4,4,5,4 %N A329231 The maximum number of times one reaches a single position during the grasshopper procedure. %C A329231 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.) %C A329231 a(n) >= (A329230(n)-1)/(n-1). %C A329231 Least values of n such that a(n) = 1, 2, 3, etc are 1, 6, 5, 9, 17, 49, 74, 198, 688, 1745 etc. %C A329231 Conjecture: a(n) = 1 if and only if n = 3, n = 7, or n = 2^k for some k. %C A329231 Conjecture: The largest values of n for which a(n) = 2, 3, 4, 5 respectively are n = 18, 68, 381, 1972. %C A329231 If the second conjecture is true, then 2, 3, 4, and 5 appear 2, 19, 87, and 313 times respectively. %C A329231 Conjecture: Every integer greater than 1 appears in this sequence a finite number of times. %H A329231 Peter Kagey, <a href="/A329231/b329231.txt">Table of n, a(n) for n = 1..2048</a> %H A329231 Mathematics Stack Exchange User Vepir, <a href="https://math.stackexchange.com/q/3418970/121988">Grasshopper jumping on circles</a> %Y A329231 Cf. A282442, A329230, A329232, A329233. %K A329231 nonn,walk %O A329231 1,5 %A A329231 _Peter Kagey_, Nov 08 2019