A266805 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones.

-14, -90, -722, -4830, -33554, -228954, -1572110, -10768122, -73825010, -505954014, -3467991794, -23769625530, -162920337422, -1116670248090, -7653777913874, -52459758093534, -359564573392850, -2464492138756122, -16891880703949070, -115778671987640634

Offset: 0

Author

Clark Kimberling, Jan 09 2016

Keywords

• easy
• sign

Comments

See A265762 for a guide to related sequences.

Examples

			Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(6),1,1,1,...] has p(0,x) = 19-14x-13x^2+2x^3+x^4, so a(0) = -14;
[1,sqrt(6),1,1,1,...] has p(1,x) = 19-90x+143x^2-90x^3+19x^4, so a(1) = -90;
[1,1,sqrt(6),1,1,1...] has p(2,x) = 361-722x+527x^2-166x^3+19x^4, so a(2) = -722.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Crossrefs

Cf. A265762, A266804, A266806, A266807.

Programs

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266804 *)
Coefficient[t, x, 1];  (* A266805 *)
Coefficient[t, x, 2];  (* A266806 *)
Coefficient[t, x, 3];  (* A266807 *)
Coefficient[t, x, 4];  (* A266804 *)

Formula

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

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)).