A369379 Number of Dabbaghian-Wu pandiagonal Latin squares of order 2n+1 with the first row in order.

1, 0, 0, 4, 0, 0, 72, 0, 0, 108, 0, 0, 4, 0, 0, 180, 0, 3, 216, 0, 0, 252, 0, 0, 264, 0, 0, 0, 0, 0, 360, 0, 5, 396, 0, 0, 432, 0, 0, 468, 0, 0, 0, 0, 0, 868, 0, 5, 576, 0

Offset

1,4

Comments

A pandiagonal Latin square is a Latin square in which the diagonal, antidiagonal and all broken diagonals and antidiagonals are transversals.

A Dabbaghian-Wu pandiagonal Latin square (see A368027) is a special type of pandiagonal Latin square (see A342306). Such squares are constructed from cyclic diagonal Latin squares (see A338562) for prime orders n=6k+1 (see Dabbaghian and Wu article) using a polynomial algorithm based on permutation of some values in Latin square. For other orders (25, 35, 49, ...) this algorithm also ensures correct pandiagonal Latin squares.

Links

Table of n, a(n) for n=1..50.

Vahid Dabbaghian and Tiankuang Wu, Constructing non-cyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms, Vol. 30, 2015, pp. 70-77, doi: 10.1016/j.jda.2014.12.001.

Index entries for sequences related to Latin squares and rectangles.

Example

n=13=6*2+1 (prime order):
.
 0  1  2  3  4  5  6  7  8  9 10 11 12
 2  3  0  1 11 12  8  4 10  7  5  6  9
 4 10 11  2  8  1  3  0 12  6  9  7  5
11  5  9  7 10  0 12  1  3  2  8  4  6
 8  7 10  5  9  6 11  2  0  4  3 12  1
12  0  4  6  7  2  9 10  5 11  1  8  3
 1  6 12  8  3  4  5 11  9 10  7  2  0
 9  2  3  4 12  8  1  6  7  5  0 10 11
10 11  5  0  1  3  7  8  4 12  6  9  2
 5  9  1 11  2 10  0 12  6  8  4  3  7
 6  8  7 10  0 11  2  9  1  3 12  5  4
 7  4  6 12  5  9 10  3  2  0 11  1  8
 3 12  8  9  6  7  4  5 11  1  2  0 10
.
n=19=6*3+1 (prime order):
.
 0  1  2  3  4  5  6  7  8  9 10 11 12
 2  3  0  1 11 12  8  4 10  7  5  6  9
 4 10 11  2  8  1  3  0 12  6  9  7  5
11  5  9  7 10  0 12  1  3  2  8  4  6
 8  7 10  5  9  6 11  2  0  4  3 12  1
12  0  4  6  7  2  9 10  5 11  1  8  3
 1  6 12  8  3  4  5 11  9 10  7  2  0
 9  2  3  4 12  8  1  6  7  5  0 10 11
10 11  5  0  1  3  7  8  4 12  6  9  2
 5  9  1 11  2 10  0 12  6  8  4  3  7
 6  8  7 10  0 11  2  9  1  3 12  5  4
 7  4  6 12  5  9 10  3  2  0 11  1  8
 3 12  8  9  6  7  4  5 11  1  2  0 10
.
n=25=6*4+1 (nonprime order):
.
 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
 3  4 15  6  7  8  9  5 11 12 13 14  0 16 17 18 19 10 21 22 23 24 20  1  2
 6  7  8  9 10 11 12 13 14 15 16 17 18 19  0 21 22 23 24  5  1  2  3  4 20
 9  5 11 12 13 14 10 16 17 18 19 20 21 22 23 24  0  1  2  3  4 15  6  7  8
12 13 14  0 16 17 18 19 10 21 22 23 24  5  1  2  3  4 20  6  7  8  9 15 11
15 16 17 18 19 20 21 22 23 24  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14
18 19 10 21 22 23 24 20  1  2  3  4 15  6  7  8  9  5 11 12 13 14  0 16 17
21 22 23 24  5  1  2  3  4 20  6  7  8  9 10 11 12 13 14 15 16 17 18 19  0
24  0  1  2  3  4 15  6  7  8  9  5 11 12 13 14 10 16 17 18 19 20 21 22 23
 2  3  4 20  6  7  8  9 15 11 12 13 14  0 16 17 18 19 10 21 22 23 24  5  1
 5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  0  1  2  3  4
 8  9  5 11 12 13 14  0 16 17 18 19 10 21 22 23 24 20  1  2  3  4 15  6  7
11 12 13 14 15 16 17 18 19  0 21 22 23 24  5  1  2  3  4 20  6  7  8  9 10
14 10 16 17 18 19 20 21 22 23 24  0  1  2  3  4 15  6  7  8  9  5 11 12 13
17 18 19 10 21 22 23 24  5  1  2  3  4 20  6  7  8  9 15 11 12 13 14  0 16
20 21 22 23 24  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
23 24 20  1  2  3  4 15  6  7  8  9  5 11 12 13 14  0 16 17 18 19 10 21 22
 1  2  3  4 20  6  7  8  9 10 11 12 13 14 15 16 17 18 19  0 21 22 23 24  5
 4 15  6  7  8  9  5 11 12 13 14 10 16 17 18 19 20 21 22 23 24  0  1  2  3
 7  8  9 15 11 12 13 14  0 16 17 18 19 10 21 22 23 24  5  1  2  3  4 20  6
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  0  1  2  3  4  5  6  7  8  9
13 14  0 16 17 18 19 10 21 22 23 24 20  1  2  3  4 15  6  7  8  9  5 11 12
16 17 18 19  0 21 22 23 24  5  1  2  3  4 20  6  7  8  9 10 11 12 13 14 15
19 20 21 22 23 24  0  1  2  3  4 15  6  7  8  9  5 11 12 13 14 10 16 17 18
22 23 24  5  1  2  3  4 20  6  7  8  9 15 11 12 13 14  0 16 17 18 19 10 21

Crossrefs

Cf. A338562, A342306, A368027, A369380.

Sequence in context: A192057 A054376 A351156 * A358292 A071608 A013451

Adjacent sequences: A369376 A369377 A369378 * A369380 A369381 A369382

Keyword

nonn,more

Author

Eduard I. Vatutin, Jan 22 2024

Status

approved