A383036 The determinant of the matrix representing a totally anti-symmetric quasigroup of order 2*n+1.

0, 9, 1250, 352947, 172186884, 129687123005, 139788510734886, 204350482177734375, 389289535005334947848, 937146152681201173795569, 2782184294469515486371964010, 9986310782535957929474146174619, 42632564145606011152267456054687500, 213501642487388555901009081409220318757

Offset

0,2

Comments

A totally antisymmetric quasigroup of order 2*n+1 is constructed in a way such that M[i][j] != M[j][i] for i!=j with m = 2*n+1, k = 2 and M[j][i] = k*(j-i) mod m for 0 <= j,i < m.

For any k != 0 mod m the resulting matrix M has the same determinant for each n.

Also the resulting matrix M is circulant and a Latin square.

Links

Paolo Xausa, Table of n, a(n) for n = 0..100

H. Michael Damm, Totally anti-symmetric quasigroups for all orders n not equal to 2 or 6, Discrete Math., 307:6 (2007), 715-729.

Wikipedia, Quasigroups

Formula

a(n) = n*(2*n+1)^(2*n) = A081131(2*n+1).

Example

For n = 1, a(1) = 9 because:
The resulting totally anti-symetric quasigroup has a matrix:
with k = 1:
  0, 1, 2,
  2, 0, 1,
  1, 2, 0
which has a determinant: 9.
with k = 2:
  0, 2, 1,
  1, 0, 2,
  2, 1, 0
has also the same determinant 9.

Mathematica

A383036[n_] := n*(2*n+1)^(2*n); Array[A383036, 15, 0] (* Paolo Xausa, May 28 2025 *)

Crossrefs

Cf. A005408, A081131, A375584.

Sequence in context: A084149 A202981 A276823 * A020261 A266864 A076442

Adjacent sequences: A383033 A383034 A383035 * A383037 A383038 A383039

Keyword

nonn,easy

Author

Darío Clavijo, May 21 2025

Status

approved