Minimum number of loops in a diagonal Latin square of order n, https://oeis.org/A307166

n=1, a(1)=1
Announcement: -, Eduard I. Vatutin, Mar 27 2019
Article: Vatutin E., Belyshev A., Nikitina N., Manzuk M. Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10 // Communications in Computer and Information Science. Vol. 1304. Springer, 2020. pp. 127-146. DOI: 10.1007/978-3-030-66895-2_9
Way of finding: brute force
0

n=2, a(2)=0
-

n=3, a(3)=0
-

n=4, a(4)=12
Announcement: -, Eduard I. Vatutin, Mar 27 2019
Article: Vatutin E., Belyshev A., Nikitina N., Manzuk M. Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10 // Communications in Computer and Information Science. Vol. 1304. Springer, 2020. pp. 127-146. DOI: 10.1007/978-3-030-66895-2_9
Way of finding: brute force
0 1 2 3
3 2 1 0
1 0 3 2
2 3 0 1

n=5, a(5)=10
Announcement: -, Eduard I. Vatutin, Mar 27 2019
Article: Vatutin E., Belyshev A., Nikitina N., Manzuk M. Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10 // Communications in Computer and Information Science. Vol. 1304. Springer, 2020. pp. 127-146. DOI: 10.1007/978-3-030-66895-2_9
Way of finding: brute force
0 1 2 3 4
4 2 3 0 1
3 4 1 2 0
1 3 0 4 2
2 0 4 1 3

n=6, a(6)=27
Announcement: -, Eduard I. Vatutin, Mar 27 2019
Article: Vatutin E., Belyshev A., Nikitina N., Manzuk M. Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10 // Communications in Computer and Information Science. Vol. 1304. Springer, 2020. pp. 127-146. DOI: 10.1007/978-3-030-66895-2_9
Way of finding: brute force
0 1 2 3 4 5
4 2 5 0 3 1
3 5 1 2 0 4
5 3 0 4 1 2
2 4 3 1 5 0
1 0 4 5 2 3

n=7, a(7)=21
Announcement: -, Eduard I. Vatutin, Mar 27 2019
Article: Vatutin E., Belyshev A., Nikitina N., Manzuk M. Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10 // Communications in Computer and Information Science. Vol. 1304. Springer, 2020. pp. 127-146. DOI: 10.1007/978-3-030-66895-2_9
Way of finding: brute force
0 1 2 3 4 5 6
4 2 6 0 5 1 3
3 5 1 6 0 4 2
5 6 3 4 1 2 0
6 4 5 2 3 0 1
1 3 0 5 2 6 4
2 0 4 1 6 3 5

n=8, a(8)=40
Announcement: https://vk.com/wall162891802_1320, Eduard I. Vatutin, Aug 03 2020
Article: Vatutin E., Belyshev A., Nikitina N., Manzuk M. Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10 // Communications in Computer and Information Science. Vol. 1304. Springer, 2020. pp. 127-146. DOI: 10.1007/978-3-030-66895-2_9
Way of finding: brute force using X-based fillings
0 1 2 3 4 5 6 7 
1 7 6 4 5 3 0 2 
6 3 1 5 0 2 7 4 
3 0 4 2 1 7 5 6 
7 2 5 6 3 4 1 0 
5 4 3 0 7 6 2 1 
2 5 7 1 6 0 4 3 
4 6 0 7 2 1 3 5

Aug 03 2020