A266701 Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,1/3,1,1,1,...], where 1^n means n ones.

9, 11, 5, 41, 81, 239, 599, 1595, 4149, 10889, 28481, 74591, 195255, 511211, 1338341, 3503849, 9173169, 24015695, 62873879, 164605979, 430944021, 1128226121, 2953734305, 7732976831, 20245196151, 53002611659, 138762638789, 363285304745, 951093275409

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.: (9 - 7 x - 35 x^2 + 18 x^3)/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(-37*(-2)^n-2*(3-sqrt(5))^n*(2+3*sqrt(5))+(3+sqrt(5))^n*(-4+6*sqrt(5))))/5. - 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) a(n) = round((2^(-n)*(-37*(-2)^n-2*(3-sqrt(5))^n*(2+3*sqrt(5))+(3+sqrt(5))^n*(-4+6*sqrt(5))))/5) \\ Colin Barker, Sep 29 2016
(PARI) Vec((9-7*x-35*x^2+18*x^3)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016

Crossrefs

Cf. A265762, A266702.

Sequence in context: A165254 A058069 A266703 * A172283 A172185 A098728

Adjacent sequences: A266698 A266699 A266700 * A266702 A266703 A266704

Keyword

nonn,easy

Author

Clark Kimberling, Jan 09 2016

Extensions

Three typos in data fixed by Colin Barker, Sep 29 2016

Status

approved