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 A384424 #22 Jun 13 2025 16:44:59 %S A384424 0,0,5,8,16,24,36,44 %N A384424 The maximal possible number of 'good' steps in a Hamiltonian cycle on the n X n king's graph, as is specified in the comments. %C A384424 The cycle is drawn on an n X n square grid. Denote the geometric center of the grid by O. An edge a -> b on the cycle is 'good' if the distance from b to O is strictly less than the distance from a to O. %C A384424 The n = 8 case is Grade 9, Problem 4 of 2025 All-Russian Olympiad. %H A384424 Yifan Xie, <a href="/A384424/a384424_2.png">Illustration of a(n), n = 2..8</a>, 'good' steps are green, the others are blue and the red circles are the centers of the grids (the construction for n = 6 from Sean A. Irvine). %e A384424 Proof that a(6) <= 24: Mark the cells on a 6 X 6 grid with the following symbols: %e A384424 ABXXBA %e A384424 BXXXXB %e A384424 XXOOXX %e A384424 XXOOXX %e A384424 BXXXXB %e A384424 ABXXBA %e A384424 The 4 steps to A and the 4 steps from O must be non-'good'. For each A, the 2 steps to the neighboring B's cannot both be 'good', or they must both be A -> B. So there are at least 4 + 4 + 4 = 12 non-'good' steps, hence a(6) <= 24. %Y A384424 Cf. A308129. %K A384424 nonn,more %O A384424 1,3 %A A384424 _Yifan Xie_, May 28 2025