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A triangle sequence of determinants: a(n)=If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b(n,m)=If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M={{a(m), b(n, m)}, {a(n), b(n, n)}}; t(n,m)=Det[M].
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%I #2 Oct 12 2012 14:54:55

%S -1,-1,1,-1,1,1,-1,1,1,-1,-1,1,0,-1,1,-1,1,-1,-1,1,-1,-1,1,-1,-1,1,-1,

%T -1,-1,1,0,-1,1,0,-1,1,-1,1,1,-1,1,1,-1,1,1,-1,1,1,-1,1,1,-1,1,1,-1

%N A triangle sequence of determinants: a(n)=If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b(n,m)=If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M={{a(m), b(n, m)}, {a(n), b(n, n)}}; t(n,m)=Det[M].

%C Row sums are:{-1, 0, 1, 0, 0, -2, -3, 0, 3, 2}.

%C It took me a while to get the projection right.

%C The example three matrices are:

%C Table[M /. n -> 4, {m, 1, 3}]

%C M1={{-1, 0},

%C {-1, -1}};

%C M2={{0, 1},

%C {-1, -1}};

%C M3={{1, 0},

%C {-1, -1}};

%C Characteristic polynomials:

%C Table[CharacteristicPolynomial[M /. n -> 4, x], {m, 1, 3}];

%C {1 + 2 x + x^2, 1 + x + x^2, -1 + x^2}.

%F a(n)=If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b(n,m)=If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M={{a(m), b(n, m)}, {a(n), b(n, n)}}; t(n,m)=Det[M].

%e {-1},

%e {-1, 1},

%e {-1, 1, 1},

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

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

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

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

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

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

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

%t Clear[a, b, t, n, m] a[n_] := If[Mod[n, 3] == 0, 1, If[Mod[n, 3] == 1, -1, If[Mod[n, 3] == 2, 0]]]; b[n_, m_] := If[m < n && Mod[m, 3] == 0, 0, If[m < n && Mod[m, 3] == 1, 0, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 0, 1, If[m < n && Mod[m, 3] == 2 && Mod[n, 2] == 1, -1, If[m == n, -1, 0]]]]]; M := {{a[m], b[n, m]}, {a[n], b[n, n]}}; t[n_, m_] := Det[M]; Table[Table[t[n, m], {m, 0, n - 1}], {n, 1, 10}]; Flatten[%]

%K sign,uned

%O 1,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 10 2008