A265805 Coefficient of x in minimal polynomial of the continued fraction [1^n,5,1,1,1,...], where 1^n means n ones.

-9, -47, -105, -295, -753, -1991, -5193, -13615, -35625, -93287, -244209, -639367, -1673865, -4382255, -11472873, -30036391, -78636273, -205872455, -538981065, -1411070767, -3694231209, -9671622887, -25320637425, -66290289415, -173550230793, -454360402991

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 - 29 x + 7 x^2)/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(27*(-2)^n + 4*(3-sqrt(5))^n*(-9+5*sqrt(5)) - 4*(3+sqrt(5))^n*(9+5*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:
[5,1,1,1,1,...] = (9 + sqrt(5))/2 has p(0,x) = 19 - 9 x + x^2, so a(0) = 1;
[1,5,1,1,1,...] = (47 - sqrt(5))/38 has p(1,x) = 29 - 47 x + 19 x^2, so a(1) = 19;
[1,1,5,1,1,...] = (105 + sqrt(5))/58 has p(2,x) = 5 - 105 x + 29 x^2, so a(2) = 29.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {5}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
Coefficient[t, x, 0] (* A265804 *)
Coefficient[t, x, 1] (* A265805 *)
Coefficient[t, x, 2] (* A236804 *)
LinearRecurrence[{2, 2, -1}, {-9, -47, -105}, 30] (* Vincenzo Librandi, Jan 06 2016 *)

Prog

(Magma) I:=[-9, -47, -105]; [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) Vec((-9-29*x+7*x^2)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ Altug Alkan, Jan 07 2016

Crossrefs

Cf. A265762, A265805.

Sequence in context: A370622 A307955 A110675 * A274270 A100790 A055250

Adjacent sequences: A265802 A265803 A265804 * A265806 A265807 A265808

Keyword

sign,easy

Author

Clark Kimberling, Jan 05 2016

Status

approved