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A123973
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Sequence of tridiagonal matrices with one center zero terminal that give a triangular sequence from the characteristic polynomials based on the 3 X 3 matrix type: {{1, -1, 0}, {-1, 1, -1}, {0, -1, 0}}.
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0
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0, 0, -1, -1, -1, 1, -1, 1, 2, -1, 0, 3, 0, -3, 1, 1, 2, -5, -2, 4, -1, 1, -2, -7, 6, 5, -5, 1, 0, -5, 0, 15, -5, -9, 6, -1, -1, -3, 12, 9, -25, 1, 14, -7, 1, -1, 3, 15, -18, -29, 35, 7, -20, 8, -1, 0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,9
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COMMENTS
| Matrices: {{0}}, {{1, -1}, {-1, 0}}, {{1, -1, 0}, {-1, 1, -1}, {0, -1, 0}}, {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 1, -1}, {0, 0, -1, 0}}, {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 1, -1, 0}, {0, 0, -1, 1, -1}, {0, 0, 0, -1, 0}}, {{1, -1, 0, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {0, -1, 1, -1, 0, 0}, {0, 0, -1, 1, -1, 0}, { 0, 0, 0, -1, 1, -1}, {0, 0, 0, 0, -1, 0}} Determinants ( not all Sl(3,Z) and invertable): Table[Det[M[d]], {d, 1, 10}] {0, -1, -1, 0, 1, 1, 0, -1, -1, 0}
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FORMULA
| t(n,m,d)=If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, 0, 0]]]
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EXAMPLE
| Triangle begins:
{0},
{0, -1},
{-1, -1, 1},
{-1, 1, 2, -1},
{0, 3, 0, -3, 1},
{1, 2, -5, -2, 4, -1},
{1, -2, -7, 6, 5, -5, 1},
{0, -5, 0, 15, -5, -9, 6, -1},
{-1, -3, 12, 9, -25, 1, 14, -7, 1},
{-1, 3, 15, -18, -29, 35,7, -20, 8, -1},
{0, 7, 0, -42, 14, 63, -42, -20, 27, -9, 1}
Some of the polynomials are Steinbach.
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MATHEMATICA
| T[n_, m_, d_] := If[ n == m && n < d && m < d, 1, If[n == m - 1 || n == m + 1, -1, If[n == m == d, 0, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[M[1], Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a]
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CROSSREFS
| Sequence in context: A128179 A178780 A058558 * A098493 A058560 A131047
Adjacent sequences: A123970 A123971 A123972 * A123974 A123975 A123976
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KEYWORD
| uned,tabl,sign
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AUTHOR
| Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 30 2006
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EXTENSIONS
| Looking at the triangle suggests that the very first term should be 1, not 0. - N. J. A. Sloane (njas(AT)research.att.com), Nov 01, 2006
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