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A123126
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Absolute value of coefficient of X^2 in the characteristic polynomial of the n-th power of the matrix M={{1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0}}.
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2
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1, 1, 4, 1, 31, 22, 1, 33, 4, 141, 199, 10, 209, 113, 604, 1473, 375, 1174, 1521, 2721, 9580, 5501, 6671, 14346, 15681, 57409, 56596, 44577, 112463, 119382, 333313, 480641, 360628, 800973, 1007191, 1988362, 3628369, 3160689, 5525420, 8309793
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OFFSET
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1,3
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COMMENTS
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Also sum of successive powers of all combinations of product of three different roots of quintic pentanacci polynomial X^5-X^4-X^3-X^2-X-1 Let roots are X1,X2,X3,X4,X5 (X1 X2 X3)^n + (X1 X2 X4)^n + (X1 X2 X5)^n + ... + (X3 X4 X5)^n A074048 are opposite coefficients by X^4 of characteristic polynomials successive powers of pentanacci matrix or successive powers of sums all roots (X1)^n+(X2)^n+(X3)^n+(X4)^n+(X5)^n.
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LINKS
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Table of n, a(n) for n=1..40.
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FORMULA
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G.f.: -x*(10*x^9 +9*x^8 +16*x^7 +21*x^6 +18*x^5 -30*x^4 +4*x^3 -3*x^2 -1) / (x^10 +x^9 +2*x^8 +3*x^7 +3*x^6 -6*x^5 +x^4 -x^3 -x +1). - Colin Barker, May 16 2013
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EXAMPLE
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a(5)=31 because characteristic polynomial fifth power of pentanacci matrix M^5 is X^5 - 31X^4 + 49X^3 - 31X^2 + 9X - 1.
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MAPLE
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with(linalg): M[1]:=matrix(5, 5, [1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0]): for n from 2 to 45 do M[n]:=multiply(M[n-1], M[1]) od: seq(-coeff(charpoly(M[n], x), x, 2), n=1..45); # Emeric Deutsch
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MATHEMATICA
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f[n_] := CoefficientList[ CharacteristicPolynomial[ MatrixPower[{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, n], x], x][[3]]; Array[f, 40] (* Robert G. Wilson v *)
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CROSSREFS
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Cf. A074048, A123127.
Sequence in context: A073323 A077097 A190647 * A303277 A174501 A051142
Adjacent sequences: A123123 A123124 A123125 * A123127 A123128 A123129
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KEYWORD
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nonn
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AUTHOR
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Artur Jasinski, Sep 30 2006
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EXTENSIONS
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Edited by N. J. A. Sloane, Oct 24 2006
More terms from Emeric Deutsch and Robert G. Wilson v, Oct 24 2006
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STATUS
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approved
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