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Search: id:A123968
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| A123968 |
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Coefficient of quadratic factor of n centered 5 X 5 tridiagonal matric characteristic polynomial: p(n,x)=(x-(n-1))*(x-n)*(x-(n+1)*(x^2-2*n*x+a(n)). |
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+0
1
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| -2, 1, 6, 13, 22, 33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321, 358, 397, 438, 481, 526, 573, 622
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Chromatic polynomials are embedded in these characteristic polynomials. The quadratic is a simple beta integer sequence: Table[ NSolve[(-3 + n^2 - 2 n x + x^2) == 0, x][[2]], {n, 1, 25}] has roots: n+Sqrt[3]
Is this a shifted version of A028872? - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 18 2008
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FORMULA
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M(n)={{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, {0, 0, 0, -1, m}}; p(n,x)=Factor(CharacteristicPolynomial(M(n)))=(-1 + n - x)((n - x)(1 + n - x)(-3 + n^2 - 2 n x + x^2) a(n)=n^2-3
a(n)=2*n+a(n-1)-1 (with a(1)=-2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]
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EXAMPLE
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A quadratic set with interesting roots.
1->x^2-2*x-2
2->x^2-4*x+1
3->x^2-6*x-6
4->13 - 8 x + x^2
5->22 - 10 x + x^2
6->33 - 12 x + x^2
For n=2, a(2)=2*2-2-1=1; n=3, a(3)=2*3+1-1=6; n=4, a(4)=2*4+6-1=13 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]
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MAPLE
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with(combinat):seq(fibonacci(3, i)-4, i=1..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
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MATHEMATICA
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M[n_] := {{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, {0, 0, 0, -1, n}}; p[n_, x_] = Factor[CharacteristicPolynomial[M[n], x]] Table[ -3 + n^2, {n, 1, 25}]
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CROSSREFS
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Sequence in context: A128534 A002562 A136456 this_sequence A068797 A049951 A025263
Adjacent sequences: A123965 A123966 A123967 this_sequence A123969 A123970 A123971
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KEYWORD
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uned,probation,sign,new
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AUTHOR
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Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 29 2006
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