A266700 Coefficient of x in minimal polynomial of the continued fraction [1^n,1/2,1,1,1,...], where 1^n means n ones.

0, -10, -12, -44, -102, -280, -720, -1898, -4956, -12988, -33990, -89000, -232992, -609994, -1596972, -4180940, -10945830, -28656568, -75023856, -196415018, -514221180, -1346248540, -3524524422, -9227324744, -24157449792, -63245024650, -165577624140

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).

G.f.: (2 (-5 x + 4 x^2))/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(9*(-1)^n*2^(1+n) + (3-sqrt(5))^n*(-9+sqrt(5)) - (3+sqrt(5))^n*(9+sqrt(5))))/5. - 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) = 0;
[1,1/2,1,1,1,...] = (5 + 2*sqrt(5))/5 has p(1,x) = 1 - 10*x + 5*x^2, so a(1) = -10;
[1,1,1/2,1,1,...] = 6 - 2*sqrt(5) has p(2,x) = 16 - 12*x + x^2, so a(2) = -12.

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 *)
LinearRecurrence[{2, 2, -1}, {0, -10, -12}, 30] (* Vincenzo Librandi, Jan 06 2016 *)

Prog

(Magma) I:=[0, -10, -12]; [n le 3 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) concat(0, Vec((-10*x+8*x^2)/(1-2*x-2*x^2+x^3) + O(x^100))) \\ Altug Alkan, Jan 07 2016

Crossrefs

Cf. A265762, A266699.

Sequence in context: A267393 A349160 A248481 * A242508 A367149 A363285

Adjacent sequences: A266697 A266698 A266699 * A266701 A266702 A266703

Keyword

sign,easy

Author

Clark Kimberling, Jan 05 2016

Status

approved