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%I #18 Jan 09 2019 01:58:08
%S 1,3,27,293,3605,52327,899311,18019017
%N Number of skew-symmetric n X n matrices A = (a_ij) with entries from {-1,0,+1} such that a_wx a_yz + a_wy a_zx + a_wz a_xy = a_wx a_wy a_wz a_xy a_xz a_yz for all distinct w,x,y,z in {1..n}.
%C These matrices have several combinatorial interpretations.
%C If the condition in the definition is changed to
%C a_wx a_yz + a_wy a_xz + a_wz a_xy = a_wx a_wy a_wz a_xy a_xz a_yz,
%C which has a different second term, we get A204809.
%H Gerhard Wesp, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00402-8">A note on the spectra of certain skew-symmetric {1,0,-1}-matrices</a>, Discrete Math. 258 (2002), no. 1-3, 339-346. doi:10.1016/S0012-365X(02)00402-8.
%e a(1)=1 (the zero matrix), and a(2) = 3, a(3) = 27 (up to this point we get all skew-symmetric matrices).
%p # Maple code for the case n=4, included to clarify the definition.
%p # It gives 293 as the answer. - _N. J. A. Sloane_, Jan 19 2012
%p with(combinat); A:=Matrix(4,4): for i from 1 to 4 do A[i,i]:=0; od:
%p n:=0;
%p for a from -1 to 1 do A[1,2]:=a; A[2,1]:=-a;
%p for b from -1 to 1 do A[1,3]:=b; A[3,1]:=-b;
%p for c from -1 to 1 do A[1,4]:=c; A[4,1]:=-c;
%p for d from -1 to 1 do A[2,3]:=d; A[3,2]:=-d;
%p for e from -1 to 1 do A[2,4]:=e; A[4,2]:=-e;
%p for f from -1 to 1 do A[3,4]:=f; A[4,3]:=-f;
%p perms:=permute(4); nsw:=+1;
%p for i from 1 to 24 do
%p p:=perms[i];
%p w:=p[1]; x:=p[2]; y:=p[3]; z:=p[4];
%p star:=A[w,x]*A[y,z]+A[w,y]*A[z,x]+A[w,z]*A[x,y]-A[w,x]*A[w,y]*A[w,z]*A[x,y]*A[x,z]*A[y,z];
%p if star <> 0 then nsw:=-1; break; fi;
%p od:
%p if nsw = 1 then n:=n+1; fi;
%p od: od: od: od: od: od:
%p n;
%Y Cf. A204809.
%K nonn,more
%O 1,2
%A _N. J. A. Sloane_, Jan 19 2012
%E a(4)-a(8) computed by _R. H. Hardin_, Jan 19 2012
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