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 A355903 #61 Sep 20 2022 03:17:27 %S A355903 1,27,41,67 %N A355903 Variant of Stepping Stones problem: here the stone you place only needs to divide the sum of its 8 neighbors. %C A355903 To find a(n), start by placing n stones labeled 1 on an infinite chessboard. %C A355903 Set k=2. At step k, you must place a stone labeled k in a square where the sum of its neighbors is x*k for some x >= 1. That is, the sum of the neighbors must be an integral multiple of k. %C A355903 If there is no way to do that, this game ends, and you win k-1 dollars. If you can do it, you increment k and repeat. %C A355903 Then a(n) = maximum number of dollars you can win with optimal play for any initial placement of the n starting stones. %C A355903 The sequence was proposed by _Skylark Xentha Murphy-Davies_ on Sep 16 2022. She found lower bounds for a(2), a(3), and a(4). On Sep 17 2022, _Hugo van der Sanden_ showed that her lower bound for a(2) was the correct value, and found the values of a(3) and a(4). %C A355903 In the original version of the problem (see A337663) each stone that you place must equal the sum of its 8 neighbors. %H A355903 Skylark Xentha Murphy-Davies and Hugo van der Sanden, <a href="/A355903/a355903.png">Illustration for a(2) = 27</a> (Discovered by Skylark Xentha Murphy-Davies and proved optimal by Hugo van der Sanden.) [There are two choices for the 27 stone, indicated by dashed lines. The 27 in the top left corner is not part of the arrangement.] %H A355903 N. J. A. Sloane, Brady Haran and Pete McPartlan, <a href="https://www.youtube.com/watch?v=m4Uth-EaTZ8">Stones on an Infinite Chessboard</a>, Numberphile video (2022). %H A355903 Hugo van der Sanden, <a href="/A355903/a355903_1.jpg">Maximal examples for a(2), a(3), a(4)</a> %e A355903 Illustration for a(2) = 27 (discovered by _Skylark Xentha Murphy-Davies_ and proved optimal by _Hugo van der Sanden_): %e A355903 . . . . . . . . %e A355903 . 21 20 . . . . . %e A355903 23 13 9 18 . . . . %e A355903 . 12 5 4 . 14 25 . %e A355903 . 27 10 1 3 11 26 . %e A355903 . . . . 2 6 . 22 %e A355903 . . . 19 . 1 7 15 %e A355903 . . . . 17 16 8 . %e A355903 . . . . . . 24 . %e A355903 . . . . . . . . %Y A355903 Cf. A337663. %K A355903 nonn,bref,more %O A355903 1,2 %A A355903 _N. J. A. Sloane_, Sep 17 2022, based on emails from _Skylark Xentha Murphy-Davies_ and _Hugo van der Sanden_