(MAGMAMagma) 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
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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.
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.
Sum of successive powers of all combinations of products of three different roots of the quintic pentanacci polynomial: Let P(x) = X^5 - X^4 - X^3 - X^2 - X - 1. Let the roots of the pentanacci polynomial be and X1, X2, X3, X4, X5 then its roots. Then a(n) = (X1*X2*X3)^n + (X1* X2*X4)^n + (X1*X2*X5)^n + ... + (X3*X4*X5)^n.
a(5) = 31 because characteristic polynomial fifth power of pentanacci matrix M^5 is X^5 - 31*X^4 + 49*X^3 - 31*X^2 + 9*X - 1.
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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)
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}}.
Also sum Sum of successive powers of all combinations of product products of three different roots of the quintic pentanacci polynomial : X^5 - X^4 - X^3 - X^2 - X - 1 . Let the roots are of the pentanacci polynomial be X1, X2, X3, X4, X5 then a(n) = (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
G. C. Greubel, <a href="/A123126/b123126.txt">Table of n, a(n) for n = 1..1000</a>
G.f.: -x*(10*x^9 1 +93*x^8 +162 -4*x^7 3 +2130*x^6 +4 -18*x^5 -3021*x^4 +46 -16*x^3 7 -39*x^2 8 -1) / (x^10 +*x^9 +2*)/(1 -x -x^8 +3 +x^4 -6*x^7 5 +3*x^6 -6+3*x^5 7 +2*x^4 -8 +x^3 -9 +x +1^10). - Colin Barker, May 16 2013
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)
a(5) = 31 because characteristic polynomial fifth power of pentanacci matrix M^5 is X^5 - 31X31*X^4 + 49X49*X^3 - 31X31*X^2 + 9X 9*X - 1.
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 *)
(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)
def A123126_list(prec):
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()
a=A123126_list(40); a[1:] # G. C. Greubel, Aug 03 2021
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