A266807 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones.

2, -90, -166, -2166, -12010, -89598, -594910, -4127706, -28160326, -193357590, -1324392298, -9079876830, -62228230846, -426534794586, -2923470679270, -20037876860598, -137341361295850, -941352453457086, -6452123715212446, -44223519044857050

Offset

0,1

Comments

See A265762 for a guide to related sequences.

Links

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Formula

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) .

G.f.: (2 (-1 + 50 x - 127 x^2 - 22 x^3 + 15 x^4))/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5).

Example

Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(6),1,1,1,...] has p(0,x)=19-14x-13x^2+2x^3+x^4, so a(0) = 2;
[1,sqrt(6),1,1,1,...] has p(1,x)=19-90x+143x^2-90x^3+19x^4, so a(1) = -90;
[1,1,sqrt(6),1,1,1...] has p(2,x)=361-722x+527x^2-166x^3+19x^4, so a(2) = -166. ~

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266804 *)
Coefficient[t, x, 1];  (* A266805 *)
Coefficient[t, x, 2];  (* A266806 *)
Coefficient[t, x, 3];  (* A266807 *)
Coefficient[t, x, 4];  (* A266804 *)

Crossrefs

Cf. A265762, A266804, A266805, A266806.

Sequence in context: A309994 A339581 A256962 * A355574 A076532 A212301

Adjacent sequences: A266804 A266805 A266806 * A266808 A266809 A266810

Keyword

sign,easy

Author

Clark Kimberling, Jan 10 2016

Status

approved