A204809 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_xz + 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, 3, 27, 125, 461, 1583, 5335, 17881, 59641, 197691, 650739, 2127381, 6910853

Offset

1,2

Comments

The condition in Wesp's paper is subtly different from the condition here. He requires

a_wx a_yz + a_wy a_zx + a_wz a_xy = a_wx a_wy a_wz a_xy a_xz a_yz, which has a different second term, and produces A204821.

Links

Table of n, a(n) for n=1..13.

Gerhard Wesp, 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 125 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[x, z]+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. A204821.

Sequence in context: A267924 A241678 A063263 * A367967 A034200 A306442

Adjacent sequences: A204806 A204807 A204808 * A204810 A204811 A204812

Keyword

nonn,more

Author

N. J. A. Sloane, Jan 19 2012

Extensions

a(4)-a(6) computed by Max Alekseyev, Jan 18 2012

a(7)-a(13) computed by R. H. Hardin, Jan 20 2012

Status

approved