%I #23 Sep 08 2022 08:45:28

%S 1,1,4,1,31,22,1,33,4,141,199,10,209,113,604,1473,375,1174,1521,2721,

%T 9580,5501,6671,14346,15681,57409,56596,44577,112463,119382,333313,

%U 480641,360628,800973,1007191,1988362,3628369,3160689,5525420,8309793

%N 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}}.

%C 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.

%H G. C. Greubel, <a href="/A123126/b123126.txt">Table of n, a(n) for n = 1..1000</a>

%F 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

%e 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.

%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 45 do M[n]:=multiply(M[n-1],M[1]) od: seq(-coeff(charpoly(M[n],x),x,2),n=1..45); # _Emeric Deutsch_

%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][[3]]; Array[f, 40] (* _Robert G. Wilson v_ *)

%o (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

%o (Sage)

%o def A123126_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o 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()

%o a=A123126_list(40); a[1:] # _G. C. Greubel_, Aug 03 2021

%Y Cf. A074048, A123127.

%K nonn

%O 1,3

%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