A266699 Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,1/2,1,1,1,...], where 1^n means n ones.

4, 5, 1, 16, 29, 89, 220, 589, 1529, 4016, 10501, 27505, 71996, 188501, 493489, 1291984, 3382445, 8855369, 23183644, 60695581, 158903081, 416013680, 1089137941, 2851400161, 7465062524, 19543787429, 51166299745, 133955111824, 350699035709, 918141995321

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) for n>3.

G.f.: (4 - 3 x - 17 x^2 + 8 x^3)/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(-9*(-1)^n*2^(1+n) + (3+sqrt(5))^n*(-1+2*sqrt(5)) - (3-sqrt(5))^n*(1+2*sqrt(5))))/5 for n>0. - Colin Barker, Oct 20 2016

Example

Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[1/2,1,1,1,1,...] = sqrt(5))/2 has p(0,x) = -5 + 4 x^2, so a(0) = 1;
[1,1/2,1,1,1,...] = (5 + 2 sqrt(5))/5 has p(1,x) = 1 - 10 x + 5 x^2, so a(1) = 19;
[1,1,1/2,1,1,...] =  6 - 2 sqrt(5) has p(2,x) = 16 - 12 x + x^2, so a(2) = 29.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {1/2}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A266699 *)
Coefficient[t, x, 1] (* A266700 *)
Coefficient[t, x, 2] (* A266699 *)
Join[{4}, LinearRecurrence[{2, 2, -1}, {5, 1, 16}, 30]] (* Vincenzo Librandi, Jan 06 2016 *)

Prog

(Magma) I:=[4, 5, 1, 16]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 06 2016
(PARI) Vec((4-3*x-17*x^2+8*x^3)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ Altug Alkan, Jan 07 2016

Crossrefs

Cf. A265762, A266700.

Sequence in context: A082051 A196848 A369950 * A234937 A210590 A108446

Adjacent sequences: A266696 A266697 A266698 * A266700 A266701 A266702

Keyword

nonn,easy

Author

Clark Kimberling, Jan 05 2016

Status

approved