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A266710 Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones. 5
-1, 1, -9, 31, 311, 1889, 13599, 91519, 631721, 4318271, 29628279, 202995649, 1391561279, 9537357311, 65371447881, 448058829919, 3071050697399, 21049268992289, 144273903091551, 988867867179391, 6777801652728809, 46455742430697599, 318412398690263799 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.: (1 -6*x -x^2 -46*x^3 -301*x^4 +260*x^5 +92*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) = -1;

[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = 1;

[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) = -9.

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}, {-1, 1, -9, 31, 311, 1889, 13599, 91519}, 30] (* Harvey P. Dale, Jun 17 2016 *)

PROG

(PARI) x='x+O('x^30); Vec((1 -6*x -x^2 -46*x^3 -301*x^4 +260*x^5 +92*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:=[31, 311, 1889, 13599, 91519]; [-1, 1, -9, ] 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, A266711, A266712, A266713.

Sequence in context: A189436 A145950 A177743 * A159279 A048547 A141573

Adjacent sequences:  A266707 A266708 A266709 * A266711 A266712 A266713

KEYWORD

sign,easy

AUTHOR

Clark Kimberling, Jan 09 2016

STATUS

approved

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Last modified July 7 16:01 EDT 2020. Contains 335496 sequences. (Running on oeis4.)