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A342990
Number of horizontally or vertically semicyclic diagonal Latin squares of order 2n+1.
7
1, 0, 240, 20160, 0, 319334400, 2167003238400, 0, 2943669154922496000, 5253122016055001088000, 0, 144827547726179682893168640000, 1214667347283206181421056000000000, 0, 184737047979495031539522261089255424000000, 3555700708206908663181998415125686517760000000, 0
OFFSET
0,3
COMMENTS
Horizontally semicyclic diagonal Latin square is a square where each row r(i) is a cyclic shift of the first row r(0) by some value d(i) (see example). Vertically semicyclic diagonal Latin square is a square where each column c(i) is a cyclic shift of the first column c(0) by some value d(i). Cyclic diagonal Latin squares (see A338562) fall under the definition of vertically and horizontally semicyclic diagonal Latin squares simultaneously, in this type of squares each row r(i) is obtained from the previous one r(i-1) using cyclic shift by some value d. Definition from A343867 includes this type of squares but not only it.
LINKS
E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
FORMULA
a(n) = A071607(n) * (2*n+1)!.
a(n) = A007705(n) * (2n)!. - Eduard I. Vatutin, Mar 15 2024
EXAMPLE
Example of cyclic diagonal Latin square of order 13:
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 11 12 0 1 (d=2)
4 5 6 7 8 9 10 11 12 0 1 2 3 (d=4)
6 7 8 9 10 11 12 0 1 2 3 4 5 (d=6)
8 9 10 11 12 0 1 2 3 4 5 6 7 (d=8)
10 11 12 0 1 2 3 4 5 6 7 8 9 (d=10)
12 0 1 2 3 4 5 6 7 8 9 10 11 (d=12)
1 2 3 4 5 6 7 8 9 10 11 12 0 (d=14 == 1 (mod 13))
3 4 5 6 7 8 9 10 11 12 0 1 2 (d=16 == 3 (mod 13))
5 6 7 8 9 10 11 12 0 1 2 3 4 (d=18 == 5 (mod 13))
7 8 9 10 11 12 0 1 2 3 4 5 6 (d=20 == 7 (mod 13))
9 10 11 12 0 1 2 3 4 5 6 7 8 (d=22 == 9 (mod 13))
11 12 0 1 2 3 4 5 6 7 8 9 10 (d=24 == 11 (mod 13))
Example of horizontally semicyclic diagonal Latin square of order 13:
0 1 2 3 4 5 6 7 8 9 10 11 12
2 3 4 5 6 7 8 9 10 11 12 0 1 (d=2)
4 5 6 7 8 9 10 11 12 0 1 2 3 (d=4)
9 10 11 12 0 1 2 3 4 5 6 7 8 (d=9)
7 8 9 10 11 12 0 1 2 3 4 5 6 (d=7)
12 0 1 2 3 4 5 6 7 8 9 10 11 (d=12)
3 4 5 6 7 8 9 10 11 12 0 1 2 (d=3)
11 12 0 1 2 3 4 5 6 7 8 9 10 (d=11)
6 7 8 9 10 11 12 0 1 2 3 4 5 (d=6)
1 2 3 4 5 6 7 8 9 10 11 12 0 (d=1)
5 6 7 8 9 10 11 12 0 1 2 3 4 (d=5)
10 11 12 0 1 2 3 4 5 6 7 8 9 (d=10)
8 9 10 11 12 0 1 2 3 4 5 6 7 (d=8)
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Eduard I. Vatutin, Jan 27 2022
STATUS
approved