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 A204821 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}. 1
 1, 3, 27, 293, 3605, 52327, 899311, 18019017 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These matrices have several combinatorial interpretations. If the condition in the definition is changed to a_wx a_yz + a_wy a_xz + a_wz a_xy = a_wx a_wy a_wz a_xy a_xz a_yz, which has a different second term, we get A204809. LINKS Wesp, Gerhard, A note on the spectra of certain skew-symmetric {1,0,-1}-matrices. Discrete Math. 258 (2002), no. 1-3, 339-346. doi:10.1016/S0012-365X(02)00402-8 EXAMPLE a(1)=1 (the zero matrix), and a(2) = 3, a(3) = 27 (up to this point we get all skew-symmetric matrices). MAPLE Maple code for the case n=4, included to clarify the definition. It gives 293 as the answer. - N. J. A. Sloane, Jan 19 2012 with(combinat); A:=Matrix(4, 4): for i from 1 to 4 do A[i, i]:=0; od: n:=0; for a from -1 to 1 do A[1, 2]:=a; A[2, 1]:=-a; for b from -1 to 1 do A[1, 3]:=b; A[3, 1]:=-b; for c from -1 to 1 do A[1, 4]:=c; A[4, 1]:=-c; for d from -1 to 1 do A[2, 3]:=d; A[3, 2]:=-d; for e from -1 to 1 do A[2, 4]:=e; A[4, 2]:=-e; for f from -1 to 1 do A[3, 4]:=f; A[4, 3]:=-f; perms:=permute(4); nsw:=+1; for i from 1 to 24 do p:=perms[i]; w:=p[1]; x:=p[2]; y:=p[3]; z:=p[4]; 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]; if star <> 0 then nsw:=-1; break; fi; od: if nsw = 1 then n:=n+1; fi; od: od: od: od: od: od: n; CROSSREFS Cf. A204809. Sequence in context: A011544 A127503 A214363 * A200903 A318108 A290576 Adjacent sequences:  A204818 A204819 A204820 * A204822 A204823 A204824 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 19 2012 EXTENSIONS a(4)-a(8) computed by R. H. Hardin, Jan 19 2012. STATUS approved

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Last modified December 17 03:08 EST 2018. Contains 318192 sequences. (Running on oeis4.)