A266711 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.

-6, 2, 18, -102, -714, -4826, -33222, -227298, -1558962, -10682534, -73226346, -501882042, -3439999878, -23577981122, -161606223954, -1107664654566, -7592048797962, -52036670543258, -356664661728582, -2444615917773474, -16755646877311986, -114844911923314982

Offset

0,1

Comments

See A265762 for a guide to related sequences.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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*(3 -16*x -49*x^2 +156*x^3 +237*x^4 -280*x^5 -88*x^6 +18*x^7)/(-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(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = -6;
[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = 2;
[1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = 18.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266710 *)
Coefficient[t, x, 1];  (* A266711 *)
Coefficient[t, x, 2];  (* A266712 *)
Coefficient[t, x, 3];  (* A266713 *)
Coefficient[t, x, 4];  (* A266710 *)
LinearRecurrence[{5, 15, -15, -5, 1}, {-6, 2, 18, -102, -714, -4826, -33222, -227298}, 30] (* G. C. Greubel, Jan 26 2018 *)

Prog

(PARI) x='x+O('x^30); Vec(2*(3 -16*x -49*x^2 +156*x^3 +237*x^4 -280*x^5 -88*x^6 +18*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5)) \\ G. C. Greubel, Jan 26 2018
(Magma) I:=[-102, -714, -4826, -33222, -227298]; [-6, 2, 18] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 26 2018

Crossrefs

Cf. A265762, A266710, A266712, A266713.

Sequence in context: A228692 A368521 A213771 * A248269 A081778 A055943

Adjacent sequences: A266708 A266709 A266710 * A266712 A266713 A266714

Keyword

sign,easy

Author

Clark Kimberling, Jan 09 2016

Status

approved