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Search: id:A139344
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| A139344 |
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A triangular sequence of polynomials of the coefficients of the characteristic polynomials of the Sylvester resultant matrices of the Bonacci polynomials: example matrix:for x^2-x-1 and x-1; {{1, -1, -1}, {1, -1, 0}, {0, 1, -1}}. |
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| -1, 0, -1, -1, 1, 3, 4, 1, -1, -1, -1, -10, -8, 0, 3, 3, -1, -1, 1, 25, 13, -4, -5, -1, -2, 6, -1, -1, -1, -56, -19, 12, 6, -4, -3, 7, -13, 10, -1, -1, 1, 119, 26, -25, -3, 12, 5, -5, -18, 34, -32, 15, -1, -1, -1, -246, -34, 44, -8, -22, 0, 10, 7, 25, -81, 93, -61, 21, -1, -1, 1, 501, 43, -70, 32, 30, -16, -18, 3, 5, -48, 166, -242, 200
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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Row sums are:
{-3, 7, -15, 31, -63, 127, -255, 511, -1023, 2047}
If the polynomials have a common factor the determinant of the matrix is zero.
I use the matrix making software from the MathWorld page.
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REFERENCES
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Weisstein, Eric W. "Sylvester Matrix." http://mathworld.wolfram.com/SylvesterMatrix.html
Blackmore, D. and Kappraff, J. "Phyllotaxis and Toral Dynamical Systems." ZAMM (1995).
Brendan Hassett, Introduction to algebraic Geometry,Cambridge University Press. New York,2007, page 75
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FORMULA
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p(x,n)=Sum(x^i,{i,0,n-1); M(n)=SylvesterMatix( p(x,n),p(x,n-1); out_n,m=Coefficients(Characteristicpolynomial(M(n))).
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EXAMPLE
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{-1, 0, -1, -1},
{1, 3, 4, 1, -1, -1},
{-1, -10, -8, 0, 3, 3, -1, -1},
{1, 25,13, -4, -5, -1, -2,6, -1, -1},
{-1, -56, -19, 12, 6, -4, -3, 7, -13, 10, -1, -1},
{1, 119, 26, -25, -3, 12, 5, -5, -18, 34, -32, 15, -1, -1},
{-1, -246, -34, 44, -8, -22, 0, 10, 7, 25, -81, 93, -61, 21, -1, -1},
{1, 501, 43, -70, 32, 30, -16, -18, 3, 5, -48, 166, -242, 200, -102, 28, -1, -1},
{-1, -1012, -53,104, -75, -28, 46, 20, -21, -20, 9,107, -348, 572, -574, 374, -157, 36, -1, -1},
{1, 2035, 64, -147,144, 3, -89, -2, 51, 19, -14, -29, -187, 735, -1314, 1502, -1177, 637, -228, 45, -1, -1}
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MATHEMATICA
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Clear[p, x] SylvesterMatrix1[poly1_, poly2_, var_] := Function[{coeffs1, coeffs2}, With[ {l1 = Length[coeffs1], l2 = Length[coeffs2]}, Join[ NestList[RotateRight, PadRight[coeffs1, l1 + l2 - 2], l2 - 2], NestList[RotateRight, PadRight[coeffs2, l1 + l2 - 2], l1 - 2] ] ] ][ Reverse[CoefficientList[poly1, var]], Reverse[CoefficientList[poly2, var]] ] p[x_, n_] := p[x.n] = x^n - Sum[x^i, {i, 0, n - 1}]; Table[SylvesterMatrix1[p[x, n], p[x, n - 1], x], {n, 2, 11}]; Table[Det[SylvesterMatrix1[p[x, n], p[x, n - 1], x]], {n, 2, 11}]; Table[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], {n, 2, 11}]; a = Table[CoefficientList[CharacteristicPolynomial[SylvesterMatrix1[p[x, n], p[x, n - 1], x], x], x], {n, 2, 11}]; Flatten[a]
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CROSSREFS
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Sequence in context: A102222 A084301 A058022 this_sequence A137925 A131107 A046547
Adjacent sequences: A139341 A139342 A139343 this_sequence A139345 A139346 A139347
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KEYWORD
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tabf,uned,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Jun 08 2008
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