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Search: id:A123966
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| A123966 |
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Jacobi tridiagonal matrices with 4 on the main diagonal used to get characteristic polynomials and a triangular sequence. |
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2
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| 4, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Matrices: {{4}}, {{4, -1}, {-1, 4}}, {{4, -1, 0}, {-1, 4, -1}, {0, -1, 4}}, {{4, -1, 0, 0}. {-1, 4, -1, 0}, {0, -1, 4, -1}, {0, 0, -1, 4}}, {{4, -1, 0, 0, 0}, {-1, 4, -1, 0, 0}, {0, -1, 4, -1, 0}, {0, 0, -1, 4, -1}, {0, 0, 0, -1, 4}}, {{4, -1, 0, 0, 0, 0}, {-1, 4, -1, 0, 0, 0}, {0, -1, 4, -1, 0, 0}, {0, 0, -1, 4, -1, 0}, {0, 0, 0, -1, 4, -1}, {0, 0, 0, 0, -1, 4}}
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REFERENCES
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Eric Weisstein's World of Mathematics, "Tridiagonal Matrix." http://mathworld.wolfram.com/TridiagonalMatrix.html
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FORMULA
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a(n,m)=If[ n == m, 4, If[n == m - 1 || n == m + 1, -1, 0]]] p(n,x)=CharacteristicPolynomial(a(n,m)) p(n,x)->t(n,m)
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EXAMPLE
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Polynomials:
4
4 - x,
15 - 8 x + x^2,
56 - 46 x + 12 x^2 - x^3,
209 - 232 x + 93 x^2 - 16 x^3 + x^4,
780 - 1091 x + 592 x^2 - 156 x^3 + 20 x^4 - x^5,
2911 - 4912 x + 3366 x^2 - 1200 x^3 + 235 x^4 - 24 x^5 + x^6
Triangular sequence:
{4},
{4, -1},
{15, -8, 1},
{56, -46,12, -1},
{209, -232, 93, -16, 1},
{780, -1091, 592, -156, 20, -1},
{2911, -4912, 3366, -1200, 235, -24, 1},
{10864, -21468, 17784, -8010, 2120, -330, 28, -1}
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MATHEMATICA
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Clear[M, T, d, a, x]; T[n_, m_] = If[ n == m, 4, 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] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]
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CROSSREFS
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Cf. A123343.
Sequence in context: A102323 A145902 A124028 this_sequence A079507 A098364 A116866
Adjacent sequences: A123963 A123964 A123965 this_sequence A123967 A123968 A123969
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KEYWORD
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uned,probation,tabl,sign
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 28 2006
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EXTENSIONS
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Looking at the triangle suggests that the very first term should be 1, not 4. - N. J. A. Sloane (njas(AT)research.att.com), Nov 01, 2006
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