A366828 Number of linearly independent solutions to the neighbor sum problem on a 4-dimensional chessboard of length (3n-1).

0, 4, 0, 88, 24, 4, 0, 136, 0, 220, 0, 88, 48, 52, 24, 136

Offset

1,2

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 4 dimensions.

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.

Links

Table of n, a(n) for n=1..16.

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

Example

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.

Crossrefs

Cf. A366410.

Sequence in context: A373766 A358294 A298616 * A337112 A357560 A013037

Adjacent sequences: A366825 A366826 A366827 * A366829 A366830 A366831

Keyword

nonn,more

Author

Sayan Dutta, Oct 25 2023

Status

approved