A266805 - OEIS

%I #4 Jan 09 2016 19:59:27

%S -14,-90,-722,-4830,-33554,-228954,-1572110,-10768122,-73825010,

%T -505954014,-3467991794,-23769625530,-162920337422,-1116670248090,

%U -7653777913874,-52459758093534,-359564573392850,-2464492138756122,-16891880703949070,-115778671987640634

%N Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones.

%C See A265762 for a guide to related sequences.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).

%F a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) .

%F G.f.:  -((2 (-7 - 10 x - 31 x^2 - 40 x^3 + 3 x^4))/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5)).

%e Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:

%e [sqrt(6),1,1,1,...] has p(0,x) = 19-14x-13x^2+2x^3+x^4, so a(0) = -14;

%e [1,sqrt(6),1,1,1,...] has p(1,x) = 19-90x+143x^2-90x^3+19x^4, so a(1) = -90;

%e [1,1,sqrt(6),1,1,1...] has p(2,x) = 361-722x+527x^2-166x^3+19x^4, so a(2) = -722.

%t u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}];

%t f[n_] := FromContinuedFraction[t[n]];

%t t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];

%t Coefficient[t, x, 0] ; (* A266804 *)

%t Coefficient[t, x, 1];  (* A266805 *)

%t Coefficient[t, x, 2];  (* A266806 *)

%t Coefficient[t, x, 3];  (* A266807 *)

%t Coefficient[t, x, 4];  (* A266804 *)

%Y Cf. A265762, A266804, A266806, A266807.

%K easy,sign

%O 0,1

%A _Clark Kimberling_, Jan 09 2016