A266705 Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,sqrt(5),1,1,1,...], where 1^n means n ones.

1, 11, 11, 45, 101, 281, 719, 1899, 4955, 12989, 33989, 89001, 232991, 609995, 1596971, 4180941, 10945829, 28656569, 75023855, 196415019, 514221179, 1346248541, 3524524421, 9227324745, 24157449791, 63245024651, 165577624139, 433487847789, 1134885919205

Offset

0,2

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) for n > 3.

G.f.: (1 + 9*x - 13*x^2 + 2*x^3)/(1 - 2*x - 2*x^2 + x^3).

a(n) = (2^(-n)*(-23*(-2)^n-(3-sqrt(5))^n*(-9+sqrt(5))+(3+sqrt(5))^n*(9+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:
[sqrt(5),1,1,1,1,...] = (-1+3*sqrt(5))/2 has p(0,x)=-11+x+x^2, so a(0) = 1;
[1,sqrt(5),1,1,1,...] = (23+3*sqrt(5))/22 has p(1,x)=11-23x+11x^2, so a(1) = 11;
[1,1,sqrt(5),1,1,...] = (45-3*sqrt(5))/22 has p(2,x)=45-45x+11x^2, so a(2) = 11.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[5]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A266705 *)
Coefficient[t, x, 1] (* A266706 *)
Coefficient[t, x, 2] (* A266705 *)

Prog

(PARI) Vec((1+9*x-13*x^2+2*x^3)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016

Crossrefs

Cf. A265762, A266706.

Sequence in context: A172507 A089766 A077699 * A228768 A003876 A014461

Adjacent sequences: A266702 A266703 A266704 * A266706 A266707 A266708

Keyword

nonn,easy

Author

Clark Kimberling, Jan 09 2016

Status

approved