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a(n) is the trace of the matrix f(X + n*f(X))/f(X), where X is the 2 X 2 matrix [-13, 17; 31, 97] and f(x) = x^3 - 5*x + 67.
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%I #24 Mar 14 2024 09:03:52

%S 1099258818702,8792791182238,29674231047422,70337212371066,

%T 137375369109982,237382335220982,376951744660878,562677231386482,

%U 801152429354606,1098970972522062,1462726494845662,1899012630282218,2414423012788542,3015551276321446,3708991054837742

%N a(n) is the trace of the matrix f(X + n*f(X))/f(X), where X is the 2 X 2 matrix [-13, 17; 31, 97] and f(x) = x^3 - 5*x + 67.

%C Old name was: As mentioned in the short description (cf. A165806 & A165808) polynomials have the property: f(x + k*f(x)) is congruent to 0 mod(f(x)). This is true even if the variable is a square matrix. For this sequence let X be a 2x2 matrix (X belongs to Z): col1:-13, 31;col2: 17, 97. Let the polynomial be X^3 -5X + 67. The present sequence is a sequence of traces of the matrices resulting from the division of f(X + k*f(X))/f(X). Here k belongs to N.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: 2*(549309615337*x^3+2197877953721*x^2+549629409347*x+1)/(x-1)^4. - _Alois P. Heinz_, Mar 13 2024

%p with(LinearAlgebra):

%p f:= x-> x^3-5*x+67:

%p a:= n-> (X-> Trace(f(X+n*f(X)).f(X)^(-1)))(<<-13|17>, <31|97>>):

%p seq(a(n), n=1..15); # _Alois P. Heinz_, Mar 13 2024

%Y Cf. A165806, A165808.

%K nonn,easy,less

%O 1,1

%A _A.K. Devaraj_, Oct 03 2009

%E a(5)-a(15) added and edited by _Alois P. Heinz_, Mar 13 2024