A368027 Number of Dabbaghian-Wu pandiagonal Latin squares of order 2n+1.

1, 0, 0, 24, 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 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.

Eduard I. Vatutin, About the Dabbaghian-Wu pandiagonal Latin squares for non-prime orders (in Russian).

Index entries for sequences related to Latin squares and rectangles.

Example

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

Crossrefs

Cf. A338562, A369379, A369380, A342306.

Sequence in context: A023923 A267334 A202184 * A357967 A353227 A075406

Adjacent sequences: A368024 A368025 A368026 * A368028 A368029 A368030

Keyword

nonn,more

Author

Eduard I. Vatutin, Dec 16 2023

Status

approved