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Number of invertible (-1,0,1) n X n matrices having (Tij = -Tji; i<j) such that all T^k (k= 1..12) are also (-1,0,1) matrices.
2

%I #7 Mar 30 2012 18:37:42

%S 2,14,92,796,7672,83944

%N Number of invertible (-1,0,1) n X n matrices having (Tij = -Tji; i<j) such that all T^k (k= 1..12) are also (-1,0,1) matrices.

%C The matrix powers T^k reach identity I for k a divisor of 12. All T^k are invertible (-1,0,1)-matrices with determinant +/-1. The matrix |Tij| is symmetric. The matrices T are "pseudo-anti-symmetric" (that is Tij=-Tji except for the main diagonal, or, equivalently, the sum of an anti-symmetric and a diagonal matrix). Their eigenvalues belong to {-1, -I, I, 1, -(-1)^(1/3), (-1)^(1/3), -(-1)^(2/3), (-1)^(2/3)}.

%e {{1,-1,0,0,0},{1,0,0,0,0},{0,0,0,-1,0},{0,0,1,1,0},{0,0,0,0,-1}}

%e qualifies since its powers are:

%e {{0,-1,0,0,0},{1,-1,0,0,0},{0,0,-1,-1,0},{0,0,1,0,0},{0,0,0,0,1}},

%e {{-1,0,0,0,0},{0,-1,0,0,0},{0,0,-1,0,0},{0,0,0,-1,0},{0,0,0,0,-1}},

%e {{-1,1,0,0,0},{-1,0,0,0,0},{0,0,0,1,0},{0,0,-1,-1,0},{0,0,0,0,1}},

%e {{0,1,0,0,0},{-1,1,0,0,0},{0,0,1,1,0},{0,0,-1,0,0},{0,0,0,0,-1}},

%e {{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0},{0,0,0,0,1}}.

%t triamatsig[li_List] := Block[{len=Sqrt[8Length[li]+1]/2-1/2}, If[IntegerQ[len], (Part[li, # ]&/@ Table[If[j>i, j(j-1)/2+i, i(i-1)/2+j], {i, len}, {j, len}])Table[If[j>i, -1, 1], {i, len}, {j, len}], li]]; n=4; it=triamatsig/@(-1+IntegerDigits[Range[0, -1+3^(n(n+1)/2)], 3, n(n+1)/2]); result4=Cases[it, (q_?MatrixQ)/; Det[q]=!=0 && And@@ Table[Union[Flatten[{MatrixPower[q, k], {-1, 0, 1}}]]==={-1, 0, 1}, {k, 25}]]

%Y Cf. A081959 A002464 A020063 A033169 A090410 A066052.

%K hard,nonn

%O 1,1

%A _Wouter Meeussen_, Aug 25 2003

%E a(6) from _Wouter Meeussen_, Nov 15 2005