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A366410
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Number of linearly independent solutions to the neighbor sum problem on a cubical (3n-1) X (3n-1) X (3n-1) chessboard.
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1
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0, 3, 0, 15, 6, 3, 0, 15, 0, 9, 0, 15, 0, 3, 6, 15
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OFFSET
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1,2
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COMMENTS
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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.
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.
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LINKS
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FORMULA
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If n is divisible by 2 or 5, then a(n) is nonzero (see Theorem 29 of Dutta et al. link).
It is conjectured that if a(n) is nonzero, then n is divisible by 2 or 5.
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EXAMPLE
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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.
+ + 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 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 - - 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 0
+ + 0 - - 0 0 0 0 0 - - 0 + + 0 0 0 0 0 + + 0 - -
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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