A267065 Coefficient of x^6 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

-18, 2498, 7790, 1588998, 47783370, 2692503902, 118222343438, 5700687414690, 265166547527598, 12504559841719910, 586597482595321322, 27572856062170808478, 1295063443722512524590, 60845329039209613792898, 2858347113718106912615150, 134283053266053897759020742

Offset

0,1

Comments

See A265762 for a guide to related sequences.

Links

G. C. Greubel, Table of n, a(n) for n = 0..595

Index entries for linear recurrences with constant coefficients, signature (34, 714, -4641, -12376, 12376, 4641, -714, -34, 1).

Formula

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).

G.f.: -((2 (-9 + 1555 x - 32145 x^2 - 271486 x^3 - 217086 x^4 + 308078 x^5 + 819 x^6 - 1961 x^7 + 15 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).

Example

Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = -18.
[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = 2498;
[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = 7790.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0];  (* A266803 *)
Coefficient[t, x, 1];  (* A266808 *)
Coefficient[t, x, 2];  (* A267061 *)
Coefficient[t, x, 3];  (* A267062 *)
Coefficient[t, x, 4];  (* A267063 *)
Coefficient[t, x, 5];  (* A267064 *)
Coefficient[t, x, 6];  (* A267065 *)
Coefficient[t, x, 7];  (* A267066 *)
Coefficient[t, x, 8];  (* A266803 *)

Crossrefs

Cf. A265762, A266803, A266808, A267061, A267062, A267063, A267064, A267066.

Sequence in context: A299696 A282830 A174637 * A356204 A243215 A162449

Adjacent sequences: A267062 A267063 A267064 * A267066 A267067 A267068

Keyword

sign,easy

Author

Clark Kimberling, Jan 10 2016

Status

approved