A266709 Coefficient of x in minimal polynomial of the continued fraction [2,1^n,2,1,1,...], where 1^n means n ones.

-7, -25, -59, -161, -415, -1093, -2855, -7481, -19579, -51265, -134207, -351365, -919879, -2408281, -6304955, -16506593, -43214815, -113137861, -296198759, -775458425, -2030176507, -5315071105, -13915036799, -36430039301, -95375081095, -249695203993

Offset

0,1

Comments

See A265762 for a guide to related sequences.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Formula

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

G.f.: (1 + 3 x - x^2)/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(9*(-2)^n+2*(3-sqrt(5))^n*(-11+5*sqrt(5))-2*(3+sqrt(5))^n*(11+5*sqrt(5))))/5. - Colin Barker, Oct 01 2016

Example

Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[2,2,1,1,1,...] = (7-sqrt(5))/2 has p(0,x) = 11 - 7 x + x^2, so a(0) = -7;
[2,1,2,1,1,1,...] = (25+sqrt(5))/10 has p(1,x) = 31 - 25 x + 5 x^2, so a(1) = -25;
[2,1,1,2,1,...] = (59-sqrt(5))/22 has p(2,x) = 79 - 59 x + 11 x^2, so a(2) = -59.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[{2}, u[n], {2}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0]  (* A236428 *)
Coefficient[t, x, 1]  (* A266709 *)
Coefficient[t, x, 2]  (* A236428 *)

Prog

(PARI) a(n) = round((2^(-n)*(9*(-2)^n+2*(3-sqrt(5))^n*(-11+5*sqrt(5))-2*(3+sqrt(5))^n*(11+5*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016
(PARI) Vec(-(7+11*x-5*x^2)/((1+x)*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Oct 01 2016

Crossrefs

Cf. A265762, A236428.

Sequence in context: A350222 A110672 A213481 * A162264 A034135 A212136

Adjacent sequences: A266706 A266707 A266708 * A266710 A266711 A266712

Keyword

sign,easy

Author

Clark Kimberling, Jan 09 2016

Status

approved