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%I #44 Apr 10 2023 15:36:54
%S -1,-3,-4,1,49,-42,-57,-31,140,497,-815,-758,311,3021,3796,-13759,
%T -7039,16086,45295,3681,-204684,-10431,365377,507914,-618001,-2642435,
%U 1427468,6214881,3341553,-16185322,-27959273,42625665,85186108,-23867663,-286766767,-193092086,854985639,900760205
%N 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}}.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A123127/b123127.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-2,-3,-3,6,-1,1,0,1,-1).
%F 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
%e 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.
%p 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
%t 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 *)
%t 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 *)
%o (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;
%o my(v=polsym(g(y),33)); vector(#v-1,n,v[n+1]) \\ _Joerg Arndt_, Nov 02 2020
%o (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
%o (Sage)
%o def A123127_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o 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()
%o a=A123127_list(40); a[1:] # _G. C. Greubel_, Aug 03 2021
%Y Cf. A074048, A123126.
%K sign,easy
%O 1,2
%A _Artur Jasinski_, Sep 30 2006
%E Edited by _N. J. A. Sloane_, Oct 24 2006
%E More terms from _Emeric Deutsch_ and _Robert G. Wilson v_, Oct 24 2006
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