A266801 Coefficient of x^2 in the minimal polynomial of the continued fraction [1^n,sqrt(3),1,1,...], where 1^n means n ones.

-7, 23, 65, 653, 3935, 28373, 190793, 1317335, 9003953, 61779965, 423273503, 2901611813, 19886759705, 136308977303, 934267517345, 6403586065133, 43890776239583, 300832001287925, 2061932830446953, 14132698865151575, 96866956468010513, 663936003630421853

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

Links

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

Crossrefs

Cf. A265762, A266799, A266800, A266802.

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.: (7 - 58 x - 55 x^2 + 122 x^3 - 5 x^4)/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5).