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A144475
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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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0
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-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, -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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Row sums are:{-1, 0, 1, 0, 0, -2, -3, 0, 3, 2}.
It took me a while to get the projection right.
The example three matrices are:
Table[M /. n -> 4, {m, 1, 3}]
M1={{-1, 0},
{-1, -1}};
M2={{0, 1},
{-1, -1}};
M3={{1, 0},
{-1, -1}};
Characteristic polynomials:
Table[CharacteristicPolynomial[M /. n -> 4, x], {m, 1, 3}];
{1 + 2 x + x^2, 1 + x + x^2, -1 + x^2}.
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FORMULA
| 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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EXAMPLE
| {-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, -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}
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MATHEMATICA
| 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[%]
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CROSSREFS
| Sequence in context: A114213 A108358 A144384 * A011758 A015088 A015166
Adjacent sequences: A144472 A144473 A144474 * A144476 A144477 A144478
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KEYWORD
| sign,uned
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 10 2008
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