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A342990 Number of horizontally or vertically semicyclic diagonal Latin squares of order 2n+1. 5
1, 0, 240, 20160, 0, 319334400, 2167003238400, 0, 2943669154922496000, 5253122016055001088000, 0, 144827547726179682893168640000, 1214667347283206181421056000000000, 0, 184737047979495031539522261089255424000000, 3555700708206908663181998415125686517760000000, 0 (list; graph; refs; listen; history; text; internal format)
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
FORMULA
a(n) = A071607(n) * (2*n+1)!
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
Sequence in context: A035841 A232428 A338562 * A342306 A205256 A110163
KEYWORD
nonn,more,hard
AUTHOR
Eduard I. Vatutin, Jan 27 2022
STATUS
approved

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Last modified March 29 08:49 EDT 2024. Contains 371268 sequences. (Running on oeis4.)