A266712 Coefficient of x^2 in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.

-5, -7, -7, 115, 607, 4615, 30427, 211687, 1442695, 9909907, 67867135, 465315847, 3188935867, 21858303175, 149816390407, 1026863749555, 7038210692767, 48240661271047, 330646286854555, 2266283690589607, 15533338646986375, 106467089195295187

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(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = -5;
[1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = -7;
[1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = -7.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Crossrefs

Cf. A265762, A266710, A266711, A266713.

Programs

Magma

I:=[115, 607, 4615, 30427, 211687]; [-5, -7, -7] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // G. C. Greubel, Jan 26 2018

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
Coefficient[t, x, 0] ; (* A266710 *)
Coefficient[t, x, 1];  (* A266711 *)
Coefficient[t, x, 2];  (* A266712 *)
Coefficient[t, x, 3];  (* A266713 *)
Coefficient[t, x, 4];  (* A266710 *)
LinearRecurrence[{5,15,-15,-5,1}, {-5, -7, -7, 115, 607, 4615, 30427, 211687}, 30] (* G. C. Greubel, Jan 26 2018 *)

PARI

x='x+O('x^30); Vec((5 -18*x -103*x^2 -180*x^3 -7*x^4 +280*x^5 +56*x^6 -14*x^7)/(-1 + 5*x +15*x^2 -15*x^3 -5*x^4 +x^5)) \\ G. C. Greubel, Jan 26 2018

Formula

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

G.f.: (5 -18*x -103*x^2 -180*x^3 -7*x^4 +280*x^5 +56*x^6 -14*x^7)/(-1 + 5*x +15*x^2 -15*x^3 -5*x^4 +x^5).