A266702 Coefficient of x in minimal polynomial of the continued fraction [1^n,1/3,1,1,1,...], where 1^n means n ones.

3, -25, -35, -117, -279, -757, -1955, -5145, -13443, -35221, -92183, -241365, -631875, -1654297, -4330979, -11338677, -29685015, -77716405, -203464163, -532676121, -1394564163, -3651016405, -9558485015, -25024438677, -65514830979, -171520054297

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.: (3 - 31 x + 9 x^2 + 6 x^3)/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(37*(-2)^n+2*(3-sqrt(5))^n*(-13+sqrt(5))-2*(3+sqrt(5))^n*(13+sqrt(5))))/5 for n>0. - Colin Barker, Sep 29 2016

Example

Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[1/3,1,1,1,...] = (-1 + 3 sqrt(5))/6 has p(0,x) = -11 + 3 x + 9 x^2, so a(0) = 9;
[1,1/3,1,1,...] = (25 + 9 sqrt(5))/22 has p(1,x) = 5 - 25 x + 11 x^2, so a(1) = 11;
[1,1,1/3,1,...] = (35 - 9 sqrt(5))/10 has p(2,x) = 41 - 35 x + 5 x^2, so a(2) = 5.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {1/3}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A266701 *)
Coefficient[t, x, 1] (* A266702 *)
Coefficient[t, x, 2] (* A266701 *)

Prog

(PARI) Vec((3-31*x+9*x^2+6*x^3)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016

Crossrefs

Cf. A265762, A266701.

Sequence in context: A304480 A129599 A042899 * A264937 A354725 A051280

Adjacent sequences: A266699 A266700 A266701 * A266703 A266704 A266705

Keyword

sign,easy

Author

Clark Kimberling, Jan 09 2016

Status

approved