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A123127 Coefficient of X^3 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}}. 3
-1, -3, -4, 1, 49, -42, -57, -31, 140, 497, -815, -758, 311, 3021, 3796, -13759, -7039, 16086, 45295, 3681, -204684, -10431, 365377, 507914, -618001, -2642435, 1427468, 6214881, 3341553, -16185322, -27959273, 42625665, 85186108, -23867663, -286766767, -193092086, 854985639, 900760205 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also sum of the successive powers of all combinations of products of two different roots of the quintic pentanacci polynomial X^5-X^4-X^3-X^2-X-1; namely (X1 X2)^n + (X1 X3)^n + (X1 X4)^n + (X1 X5)^n + (X2 X3)^n+ (X2 X4)^n + (X2 X5)^n + (X3 X4)^n + (X3 X5)^n + (X4 X5)^n, where X1,X2,X3,X4,X5 are the roots. A074048 are the coefficients, with changed signs, of X^4 in the characteristic polynomials of the successive powers of the pentanacci matrix or (X1)^n+(X2)^n+(X3)^n+(X4)^n+(X5)^n.

Let g(y) = y^10 + y^9 + 2*y^8 + 3*y^7 + 3*y^6 - 6*y^5 + y^4 - y^3 - y + 1 and {y1,...,y10} be the roots of g(y). Then a(n) = y1^n + ... + y10^n. - Kai Wang, Nov 01 2020

LINKS

Table of n, a(n) for n=1..38.

FORMULA

G.f.: -x*(10*x^9-9*x^8-7*x^6+6*x^5-30*x^4+12*x^3+9*x^2+4*x+1) / (x^10-x^9-x^7+x^6-6*x^5+3*x^4+3*x^3+2*x^2+x+1). - Colin Barker, May 16 2013

EXAMPLE

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

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 40 do M[n]:=multiply(M[n-1], M[1]) od: seq(coeff(charpoly(M[n], x), x, 3), n=1..40); # Emeric Deutsch, Oct 24 2006

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][[4]]; Array[f, 36] (* Robert G. Wilson v, Oct 24 2006 *)

PROG

(PARI) g(y) = y^10 + y^9 + 2*y^8 + 3*y^7 + 3*y^6 - 6*y^5 + y^4 - y^3 - y + 1;

my(v=polsym(g(y), 33)); vector(#v-1, n, v[n+1]) \\ Joerg Arndt, Nov 02 2020

CROSSREFS

Cf. A074048.

Sequence in context: A224069 A157783 A123951 * A167876 A241833 A332096

Adjacent sequences:  A123124 A123125 A123126 * A123128 A123129 A123130

KEYWORD

sign,easy

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 8 13:25 EST 2021. Contains 341948 sequences. (Running on oeis4.)