A266802 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(3),1,1,...], where 1^n means n ones.

2, -12, -16, -294, -1552, -11868, -78142, -543996, -3706624, -25463142, -174376288, -1195587372, -8193644926, -56162781804, -384938354032, -2638425262758, -18083987259952, -123949619666556, -849562999302334, -5822992294650972, -39911380656754528

Offset: 0

Author

Clark Kimberling, Jan 09 2016

Keywords

• sign
• easy

Comments

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

Links

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

Crossrefs

Cf. A265762, A266299, A266800, A266801.

Programs

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 *)

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 (-1 + 11 x - 7 x^2 + 2 x^3 + 6 x^4))/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5).