Search: id:A072148
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%I A072148
%S A072148 2,14,92,796,7672,83944
%N A072148 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 A072148 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 A072148 {{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 A072148 qualifies since its powers are:
%e A072148 {{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 A072148 {{-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 A072148 {{-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 A072148 {{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 A072148 {{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 A072148 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 A072148 Cf. A081959 A002464 A020063 A033169 A090410 A066052.
%Y A072148 Sequence in context: A081959 A002464 A020063 this_sequence A033169 A090410 
               A066052
%Y A072148 Adjacent sequences: A072145 A072146 A072147 this_sequence A072149 A072150 
               A072151
%K A072148 hard,nonn
%O A072148 1,1
%A A072148 Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 25 2003
%E A072148 a(6) from Wouter Meeussen (wouter.meeussen(AT)pandora.be), Nov 15 2005

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