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

%I A383036 #51 May 28 2025 09:18:26
%S A383036 0,9,1250,352947,172186884,129687123005,139788510734886,
%T A383036 204350482177734375,389289535005334947848,937146152681201173795569,
%U A383036 2782184294469515486371964010,9986310782535957929474146174619,42632564145606011152267456054687500,213501642487388555901009081409220318757
%N A383036 The determinant of the matrix representing a totally anti-symmetric quasigroup of order 2*n+1.
%C A383036 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.
%C A383036 For any k != 0 mod m the resulting matrix M has the same determinant for each n.
%C A383036 Also the resulting matrix M is circulant and a Latin square.
%H A383036 Paolo Xausa, <a href="/A383036/b383036.txt">Table of n, a(n) for n = 0..100</a>
%H A383036 H. Michael Damm, <a href="https://doi.org/10.1016/j.disc.2006.05.033">Totally anti-symmetric quasigroups for all orders n not equal to 2 or 6</a>, Discrete Math., 307:6 (2007), 715-729.
%H A383036 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quasigroup">Quasigroups</a>
%F A383036 a(n) = n*(2*n+1)^(2*n) = A081131(2*n+1).
%e A383036 For n = 1, a(1) = 9 because:
%e A383036 The resulting totally anti-symetric quasigroup has a matrix:
%e A383036 with k = 1:
%e A383036   0, 1, 2,
%e A383036   2, 0, 1,
%e A383036   1, 2, 0
%e A383036 which has a determinant: 9.
%e A383036 with k = 2:
%e A383036   0, 2, 1,
%e A383036   1, 0, 2,
%e A383036   2, 1, 0
%e A383036 has also the same determinant 9.
%t A383036 A383036[n_] := n*(2*n+1)^(2*n); Array[A383036, 15, 0] (* _Paolo Xausa_, May 28 2025 *)
%Y A383036 Cf. A005408, A081131, A375584.
%K A383036 nonn,easy
%O A383036 0,2
%A A383036 _Darío Clavijo_, May 21 2025