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
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (-1,-2,-3,-3,6,-1,1,0,1,-1).
FORMULA
G.f.: -x*(1 +4*x +9*x^2 +12*x^3 -30*x^4 +6*x^5 -7*x^6 -9*x^8 +10*x^9)/(1 +x +2*x^2 +3*x^3 +3*x^4 -6*x^5 +x^6 -x^7 -x^9 +x^10). - Colin Barker, May 16 2013
EXAMPLE
a(5) = 49 because the characteristic polynomial of fifth power of pentanacci matrix M^5 is X^5 -31*X^4 +49*X^3 -31*X^2 +9*X -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 *)
LinearRecurrence[{-1, -2, -3, -3, 6, -1, 1, 0, 1, -1}, {-1, -3, -4, 1, 49, -42, -57, -31, 140, 497}, 40] (* Harvey P. Dale, Apr 10 2023 *)
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
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( -x*(1+4*x+9*x^2 +12*x^3-30*x^4+6*x^5-7*x^6-9*x^8+10*x^9)/(1+x+2*x^2+3*x^3+3*x^4-6*x^5 +x^6 -x^7 -x^9+x^10) )); // G. C. Greubel, Aug 03 2021
(Sage)
def A123127_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( -x*(1+4*x+9*x^2+12*x^3-30*x^4+6*x^5-7*x^6-9*x^8+10*x^9)/(1+x+2*x^2 +3*x^3+3*x^4-6*x^5+x^6-x^7-x^9+x^10) ).list()
a=A123127_list(40); a[1:] # G. C. Greubel, Aug 03 2021
CROSSREFS
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