Maximum number of intercalates in a diagonal Latin square of order n, https://oeis.org/A307164

n=1, a(1)=0
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
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)=4
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 0 1 3
1 4 3 2 0
3 0 1 4 2
2 3 4 0 1

n=6, a(6)=9
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)=30
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 5 0 1 3
3 6 1 0 5 2 4
6 3 5 4 1 0 2
1 5 3 2 6 4 0
5 0 4 6 2 3 1
2 4 0 1 3 6 5

n=8, a(8)=112
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 7
4 2 1 7 0 6 5 3
5 7 3 2 6 0 4 1
7 5 6 4 3 1 2 0
6 3 7 1 5 4 0 2
1 0 4 6 2 7 3 5
3 6 5 0 7 2 1 4
2 4 0 5 1 3 7 6

n=9, a(9)=72
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 8
6 8 3 4 7 2 0 5 1
5 7 6 1 8 0 2 4 3
7 3 8 2 5 4 1 0 6
2 4 0 8 1 3 5 6 7
4 5 1 6 0 7 3 8 2
8 6 7 5 3 1 4 2 0
1 0 5 7 2 6 8 3 4
3 2 4 0 6 8 7 1 5

n=10, a(10)>=93
Announcement: https://vk.com/wall162891802_1540, Eduard I. Vatutin, Feb 02 2021
Way of finding: random search
0 8 6 5 7 1 2 9 4 3 
9 1 7 4 6 8 3 0 5 2 
4 3 2 9 5 0 1 8 6 7 
1 9 4 3 2 6 7 5 0 8 
5 2 3 0 4 9 8 1 7 6 
3 7 8 1 0 5 9 6 2 4 
8 0 5 2 3 7 6 4 9 1 
2 6 1 8 9 4 0 7 3 5 
7 4 9 6 1 3 5 2 8 0 
6 5 0 7 8 2 4 3 1 9 

n=11, a(11)>=94
Announcement: https://vk.com/wall162891802_1549, Eduard I. Vatutin, Mar 03 2021
Way of finding: random search
0 1 2 3 4 5 6 7 8 9 10 
2 3 0 1 5 6 8 10 9 4 7 
5 6 10 8 0 2 1 9 3 7 4 
10 8 7 9 6 4 3 2 1 0 5 
1 0 9 2 7 8 5 4 10 6 3 
3 9 4 10 2 1 0 6 7 5 8 
8 10 6 7 9 3 4 0 5 2 1 
9 2 1 0 8 7 10 5 4 3 6 
7 4 8 5 1 0 2 3 6 10 9 
4 7 5 6 3 10 9 1 2 8 0 
6 5 3 4 10 9 7 8 0 1 2 

n=12, a(12)>=252
Announcement: https://vk.com/wall162891802_1712, Eduard I. Vatutin, Jul 28 2021
Way of finding: random search in the neighborhood of central symmetry + combinatorial structures ODLS CFs analysis
0 1 2 3 4 5 6 7 8 9 10 11
1 2 0 4 5 3 8 6 7 11 9 10
3 5 4 0 2 1 10 9 11 7 6 8
9 10 11 6 7 8 3 4 5 0 1 2
10 11 9 7 8 6 5 3 4 2 0 1
6 8 7 9 11 10 1 0 2 4 3 5
2 0 1 5 3 4 7 8 6 10 11 9
4 3 5 1 0 2 9 11 10 6 8 7
5 4 3 2 1 0 11 10 9 8 7 6
11 9 10 8 6 7 4 5 3 1 2 0
7 6 8 10 9 11 0 2 1 3 5 4
8 7 6 11 10 9 2 1 0 5 4 3

n=13, a(13)>=156
Announcement: https://vk.com/wall162891802_1904, Eduard I. Vatutin, Jan 23 2022
Way of finding: expanding spectrum using 1 intercalate/loop neighborhoods based improvement
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 0 4 1 6 5 8 12 10 11 9 7
4 7 1 8 0 12 9 2 10 6 3 5 11
5 11 4 12 9 8 10 6 0 2 7 1 3
11 6 12 0 8 9 1 5 4 7 2 3 10
3 2 7 11 5 10 8 1 6 12 4 0 9
1 10 5 9 6 2 11 12 7 3 0 4 8
8 12 6 2 7 0 3 4 5 11 9 10 1
12 8 11 7 10 4 0 3 9 1 5 2 6
10 0 3 6 12 7 2 9 11 5 1 8 4
9 4 8 1 3 11 7 10 2 0 6 12 5
6 5 9 10 2 1 12 11 3 4 8 7 0
7 9 10 5 11 3 4 0 1 8 12 6 2

n=14, a(14)>=353
Announcement: https://vk.com/wall162891802_2475, Eduard I. Vatutin, Aug 06 2023
Way of finding: neighborhoods of different special types of DLS
0 1 2 3 4 5 6 7 8 9 10 11 12 13
12 13 4 7 2 10 5 8 3 11 6 9 0 1
7 5 10 11 12 4 13 0 9 1 2 3 8 6
8 6 13 9 10 2 1 12 11 3 4 0 7 5
11 10 5 12 6 13 4 9 0 7 1 8 3 2
5 7 0 4 3 11 12 1 2 10 9 13 6 8
13 12 11 10 9 8 7 6 5 4 3 2 1 0
2 3 8 1 7 0 9 4 13 6 12 5 10 11
3 9 6 0 5 1 2 11 12 8 13 7 4 10
1 0 9 6 11 3 8 5 10 2 7 4 13 12
4 11 1 5 0 7 3 10 6 13 8 12 2 9
9 2 12 8 13 6 10 3 7 0 5 1 11 4
6 8 3 2 1 9 0 13 4 12 11 10 5 7
10 4 7 13 8 12 11 2 1 5 0 6 9 3

Aug 06 2023