A266799 Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,sqrt(3),1,1,...], where 1^n means n ones.

1, 1, 49, 229, 1861, 12001, 84241, 572209, 3935569, 26939221, 184737301, 1265964481, 8677687969, 59476087009, 407659540081, 2794128600901, 19151272325221, 131264694791329, 899701808208049, 6166647394567441, 42266831441062801, 289701168799073461

Offset

0,3

Comments

See A265762 for a guide to related sequences.

Links

Table of n, a(n) for n=0..21.

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 + 4 x - 29 x^2 + 16 x^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) = 1;
[1,sqrt(3),1,1,1,...] has p(1,x)=1-12x+23x^2-12x^3+x^4, so a(1) = 1;
[1,1,sqrt(3),1,1,1...] has p(2,x)=49-98x+65x^2-16x^3+x^4, so a(2) = 49.

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 *)
LinearRecurrence[{5, 15, -15, -5, 1}, {1, 1, 49, 229, 1861}, 30] (* Harvey P. Dale, Oct 21 2019 *)

Crossrefs

Cf. A265762, A266800, A266801, A266802.

Sequence in context: A017150 A137880 A264538 * A211741 A211761 A322675

Adjacent sequences: A266796 A266797 A266798 * A266800 A266801 A266802

Keyword

nonn,easy

Author

Clark Kimberling, Jan 09 2016

Status

approved