A266806 Coefficient of x^2 in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones. S.

-13, 143, 527, 4859, 30119, 214847, 1450643, 10000367, 68393039, 469166939, 3214686407, 22036489343, 151033273907, 1035215971919, 7095427362959, 48632909524667, 333334588608743, 2284710128883647, 15659633909836499, 107332733533045679, 735669484346002127

Offset: 0

Author

Clark Kimberling, Jan 10 2016

Keywords

• sign
• easy

Comments

ee A265762 for a guide to related sequences.

Examples

			Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[sqrt(6),1,1,1,...] has p(0,x)=19-14x-13x^2+2x^3+x^4, so a(0) = -13;
[1,sqrt(6),1,1,1,...] has p(1,x)=19-90x+143x^2-90x^3+19x^4, so a(1) = 143;
[1,1,sqrt(6),1,1,1...] has p(2,x)=361-722x+527x^2-166x^3+19x^4, so a(2) = 527.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Crossrefs

Cf. A265762, A266804, A266805, A266807.

Programs

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266804 *)
Coefficient[t, x, 1];  (* A266805 *)
Coefficient[t, x, 2];  (* A266806 *)
Coefficient[t, x, 3];  (* A266807 *)
Coefficient[t, x, 4];  (* A266804 *)

PARI

Vec((13-208*x-7*x^2+116*x^3+x^4)/(-1+5*x+15*x^2-15*x^3-5*x^4+x^5) + O(x^200)) \\ Altug Alkan, Jan 10 2015

Formula

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) .

G.f.: (13 - 208 x - 7 x^2 + 116 x^3 + x^4)/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5).