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A123974
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Fibonacci central tridiagonal matrices as a triangular sequence from a recursive polynomial definition.
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0
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1, 1, -1, 0, -2, 1, -1, -3, 4, -1, -3, -6, 14, -7, 1, -14, -24, 72, -48, 12, -1, -109, -172, 586, -449, 143, -20, 1, -1403, -2103, 7718, -6375, 2296, -402, 33, -1, -29354, -42588, 163595, -141144, 54448, -10718, 1094, -54, 1, -996633, -1416535, 5597100, -4956116, 1990080, -418458, 47881, -2929, 88, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Matrices: {{1}}, {{1, -1}, {-1, 1}}, {{1, -1, 0}, {-1, 1, -1}, {0, -1, 2}}, {{1, -1, 0, 0}, {-1, 1, -1, 0}, {0, -1, 2, -1}, {0, 0, -1, 3}}, {{1, -1, 0, 0, 0}, {-1, 1, -1, 0, 0}, {0, -1, 2, -1, 0}, {0, 0, -1, 3, -1}, {0, 0, 0, -1, 5}}, {{1, -1, 0, 0, 0, 0}, {-1, 1, -1, 0, 0, 0}, {0, -1, 2, -1, 0, 0}, {0, 0, -1, 3, -1, 0}, {0, 0, 0, -1, 5, -1}, {0, 0, 0, 0, -1, 8}} The Dombrowski paper defines a recursive polynomial form from the tridiagonal matrices: p[1,x]=1,p[2,x]=(x-b[1])/a[1] p[n,x]=((x-b[n-1])*p[n-1,x]-a[n-2]*p[n-2,x])/a[n-1] As long as b[n-1]/a[n-1] and a[n-2]/a[n-1] behave well ( rationally or like Integers) this definition is a good recursive polynomial on a tridiagonal matrix. Here I use: a[n]=-1 and b[n]=Fibonacci[n]
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REFERENCES
| Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334
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FORMULA
| M(n,m)=If[ n == m, Fibonacci[n], If[n == m - 1 || n == m + 1, -1, 0]]
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EXAMPLE
| Triangular sequence:
{1},
{1, -1},
{0, -2, 1},
{-1, -3, 4, -1},
{-3, -6, 14, -7, 1},
{-14, -24, 72, -48, 12, -1},
{-109, -172, 586, -449, 143, -20, 1},
{-1403, -2103, 7718, -6375,2296, -402, 33, -1},
{-29354, -42588, 163595, -141144, 54448, -10718, 1094, -54, 1}
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MATHEMATICA
| T[n_, m_] := If[ n == m, Fibonacci[n], If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m], {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]
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CROSSREFS
| Sequence in context: A171881 A160449 A089940 * A193736 A056863 A120019
Adjacent sequences: A123971 A123972 A123973 * A123975 A123976 A123977
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KEYWORD
| uned,sign
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AUTHOR
| Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 30 2006
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