A002047 Number of 3 X (2n+1) zero-sum arrays with entries -n,...,0,...,n.
1, 2, 6, 28, 244, 2544, 35600, 659632, 15106128, 425802176, 14409526080, 577386122880
Offset: 0
Examples
a(2) = 6 corresponds to
..O.X.X.......X.X.O.......O.X.X.......X.O.X.......X.O.X.......X.X.O
.X.X.O.X.....X.O.X.X.....X.X.X.O.....X.X.X.O.....O.X.X.X.....O.X.X.X
X.X.X.X.O...O.X.X.X.X...X.O.X.X.X...O.X.X.X.X...X.X.X.X.O...X.X.X.O.X
.O.X.X.X.....X.X.X.O.....X.X.X.O.....X.O.X.X.....X.X.O.X.....O.X.X.X
..X.O.X.......X.O.X.......O.X.X.......X.X.O.......O.X.X.......X.X.O
The bijection with a pair of the 3 X (2n+1) zero-sum arrays:
n=1, a(1)=2 corresponds to
3 4 2 3 2 4
and mirror image 4 2 3 2 4 3
element 2 3 4 -(2n+1) --> -1 0 1
position, left element 3 1 2 -( n+1) --> 1 -1 0
position in mirror 2 3 1 -( n+1) --> 0 1 -1
------- -------
sum of column 7 7 7 -(4n+3) 0 0 0
Swapping rows 2,3 yields the other 3 X 3 zero sum array.
n=2, a(2)=6 an example and its mirror, so 2 of the 6 solutions:
5 6 7 3 4 5 3 6 4 7
mirror image 7 4 6 3 5 4 3 7 6 5
3 4 5 6 7 -(2n+1) --> -2 -1 0 1 2
4 5 1 2 3 -( n+1) --> 1 2 -2 -1 0
4 2 5 3 1 -( n+1) --> 1 -1 2 0 -2
-------------- --------------
11 11 11 11 11 -(4n+3) --> 0 0 0 0 0
Swapping rows 2,3 yields the other 3 X 5 zero sum array.
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
- B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31.
- B. T. Bennett and R. B. Potts, Arrays and brooks, J. Austral. Math. Soc., 7 (1967), 23-31. [Annotated scanned copy]
- N. J. Cavenagh and I. M. Wanless, On the number of transversals in Cayley tables of cyclic groups, Disc. Appl. Math. 158 (2010), 136-146.
- Gheorghe Coserea, Solutions for n=4.
- Gheorghe Coserea, Solutions for n=5.
- Gheorghe Coserea, MiniZinc model for generating solutions.
- Diane Donovan, Asha Rao, Elif Üsküplü, and E. Ş. Yazıcı, QC-LDPC Codes from Difference Matrices and Difference Covering Arrays, arXiv:2205.00563 [math.CO], 2022.
- A. Kotzig and P. J. Laufer, When are permutations additive?, Amer. Math. Monthly, 85 (1978), 364-365.
- A. Kotzig and P. J. Laufer, When are permutations additive?, Amer. Math. Monthly, 85 (1978), 364-365. [Annotated by C. L. Mallows, scanned copy, together with letter from C. L. Mallows and N. J. A. Sloane to A. Kotzig, Jul 25 1978]
- Wikipedia, Hexagonal chess - Gliński's hexagonal chess
Extensions
More terms from Alex Fink (a00(AT)shaw.ca), Mar 16 2005
a(10) and a(11) from Ian Wanless, Jul 30 2010, from the Cavenagh-Wanless paper
Comments