A366410 Number of linearly independent solutions to the neighbor sum problem on a cubical (3n-1) X (3n-1) X (3n-1) chessboard.

0, 3, 0, 15, 6, 3, 0, 15, 0, 9, 0, 15, 0, 3, 6, 15

Offset: 1

Author

Sayan Dutta, Oct 09 2023

Keywords

• nonn
• more

Comments

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.

Examples

			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 - -
		

Links

• Sayan Dutta, Ayanava Mandal, Sohom Gupta, and Sourin Chatterjee,Neighbour Sum Patterns: Chessboards to Toroidal Worlds, arXiv:2310.04401 [math.NT], 2023.

Formula

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.