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 A366410 #42 Oct 24 2023 15:43:03 %S A366410 0,3,0,15,6,3,0,15,0,9,0,15,0,3,6,15 %N A366410 Number of linearly independent solutions to the neighbor sum problem on a cubical (3n-1) X (3n-1) X (3n-1) chessboard. %C A366410 We say that a chessboard filled with integer entries satisfies the neighbor-sum property if the number appearing on each cell is the sum of entries in its neighboring cells, where neighbors are cells sharing a common edge or vertex. It has been proven in the paper by Dutta et. al. that an n X n chessboard satisfies this property if and only if 6 divides (n+1). The sequence at hand deals with the analogous problem in 3 dimensions. %C A366410 It has been proved that if the number of linearly independent solutions to the neighbor sum problem on a cubical n X n X n chessboard is nonzero, then 3 divides (n+1) - Theorem 28 of Dutta et. al. link. So, the sequence at hand considers a(n) = b(3*n-1) where b(n) is the number of linearly independent solutions to the neighbor sum problem on a cubical n X n X n chessboard. %H A366410 Sayan Dutta, Ayanava Mandal, Sohom Gupta, and Sourin Chatterjee,<a href="https://arxiv.org/abs/2310.04401">Neighbour Sum Patterns: Chessboards to Toroidal Worlds</a>, arXiv:2310.04401 [math.NT], 2023. %F A366410 If n is divisible by 2 or 5, then a(n) is nonzero (see Theorem 29 of Dutta et al. link). %F A366410 It is conjectured that if a(n) is nonzero, then n is divisible by 2 or 5. %e A366410 The case of n=2 corresponds to a 5 X 5 X 5 chessboard. One solution is shown below with +1 and -1 denoted just by + and - respectively. Switching coordinate axis gives two other independent solutions and so a(2) = 3. In each of these solutions a +1 (or -1) is adjacent to exactly one other and each 0 is adjacent to an equal number of +1's and -1's. %e A366410 + + 0 - - 0 0 0 0 0 - - 0 + + 0 0 0 0 0 + + 0 - - %e A366410 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %e A366410 - - 0 + + 0 0 0 0 0 + + 0 - - 0 0 0 0 0 - - 0 + + %e A366410 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %e A366410 + + 0 - - 0 0 0 0 0 - - 0 + + 0 0 0 0 0 + + 0 - - %K A366410 nonn,more %O A366410 1,2 %A A366410 _Sayan Dutta_, Oct 09 2023