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A123126 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}}. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..40.

FORMULA

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

EXAMPLE

a(5)=31 because characteristic polynomial fifth power of pentanacci matrix M^5 is X^5 - 31X^4 + 49X^3 - 31X^2 + 9X - 1.

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A074048, A123127.

Sequence in context: A073323 A077097 A190647 * A303277 A174501 A051142

Adjacent sequences:  A123123 A123124 A123125 * A123127 A123128 A123129

KEYWORD

nonn

AUTHOR

Artur Jasinski, Sep 30 2006

EXTENSIONS

Edited by N. J. A. Sloane, Oct 24 2006

More terms from Emeric Deutsch and Robert G. Wilson v, Oct 24 2006

STATUS

approved

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Last modified March 3 12:16 EST 2021. Contains 341762 sequences. (Running on oeis4.)