A266800 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(3),1,1,...], where 1^n means n ones.

-8, -12, -98, -636, -4424, -30138, -207032, -1417788, -9720866, -66619404, -456638168, -3129787002, -21452029928, -147034005996, -1007787102434, -6907472856348, -47344530365672, -324504220137018, -2224185061818776, -15244791078484764, -104489352838678178

Offset

0,1

Comments

See A265762 for a guide to related sequences.

Links

Table of n, a(n) for n=0..20.

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 (-4 + 14 x + 41 x^2 - 43 x^3 + 3 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(3),1,1,1,...] has p(0,x)=1-8x-7x^2+2x^3+x^4, so a(0) = -8;
[1,sqrt(3),1,1,1,...] has p(1,x)=1-12x+23x^2-12x^3+x^4, so a(1) = -12;
[1,1,sqrt(3),1,1,1...] has p(2,x)=49-98x+65x^2-16x^3+x^4, so a(2) = -98.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[3]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266799 *)
Coefficient[t, x, 1];  (* A266800 *)
Coefficient[t, x, 2];  (* A266801 *)
Coefficient[t, x, 3];  (* A266802 *)
Coefficient[t, x, 4];  (* A266799 *)

Crossrefs

Cf. A265762, A266799, A266801, A266802.

Sequence in context: A226259 A162466 A098664 * A230545 A180922 A083128

Adjacent sequences: A266797 A266798 A266799 * A266801 A266802 A266803

Keyword

sign,easy

Author

Clark Kimberling, Jan 09 2016

Status

approved