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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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4
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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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Let P(x) = X^5 - X^4 - X^3 - X^2 - X - 1 and X1, X2, X3, X4, X5 its roots. Then a(n) = (X1*X2*X3)^n + (X1*X2*X4)^n + (X1*X2*X5)^n + ... + (X3*X4*X5)^n.
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LINKS
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FORMULA
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G.f.: x*(1 +3*x^2 -4*x^3 +30*x^4 -18*x^5 -21*x^6 -16*x^7 -9*x^8 -10*x^9)/(1 -x -x^3 +x^4 -6*x^5 +3*x^6 +3*x^7 +2*x^8 +x^9 +x^10). - Colin Barker, May 16 2013
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EXAMPLE
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a(5) = 31 because the characteristic polynomial of M^5 is X^5 - 31*X^4 + 49*X^3 - 31*X^2 + 9*X - 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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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1 +3*x^2 -4*x^3 +30*x^4 -18*x^5 -21*x^6 -16*x^7 -9*x^8 -10*x^9)/(1 -x -x^3 +x^4 -6*x^5 +3*x^6 +3*x^7 +2*x^8 +x^9 +x^10) )); // G. C. Greubel, Aug 03 2021
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x*(1 +3*x^2 -4*x^3 +30*x^4 -18*x^5 -21*x^6 -16*x^7 -9*x^8 -10*x^9)/(1 -x -x^3 +x^4 -6*x^5 +3*x^6 +3*x^7 +2*x^8 +x^9 +x^10) ).list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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