OFFSET
1,3
COMMENTS
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
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