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.
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
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