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%I #12 Dec 31 2023 00:56:46
%S 0,4,0,88,24,4,0,136,0,220,0,88,48,52,24,136
%N Number of linearly independent solutions to the neighbor sum problem on a 4-dimensional chessboard of length (3n-1).
%C 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 4 dimensions.
%C It has been proved that if the number of linearly independent solutions to the neighbor sum problem on a 4-dimensional chessboard of length n is nonzero, then 3 divides (n+1) -- see Theorem 28 of Dutta et al. 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 4-dimensional chessboard of length n.
%H 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.
%e The case of n=2 corresponds to a 5 X 5 X 5 X 5 chessboard. Examples for the analogous problem in two dimensions are given in the paper by Dutta et al. Examples for the analogous problem in three dimensions are given in A366410.
%Y Cf. A366410.
%K nonn,more
%O 1,2
%A _Sayan Dutta_, Oct 25 2023