A267083 Coefficient of x^5 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

3, -15, -111, -1419, -32847, -549633, -10161885, -180368529, -3250331151, -58231071723, -1045558051887, -18757368165345, -336617548680381, -6040149618970929, -108387507398297007, -1944925169961946443, -34900332815651650575, -626260604480315081409

Offset

0,1

Comments

See A265762 for a guide to related sequences.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (13, 104, -260, -260, 104, 13, -1).

Formula

a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.

G.f.: -((3 (-1 + 18 x + 76 x^2 - 788 x^3 + 1992 x^4 + 2706 x^5 - 1051 x^6 - 130 x^7 + 10 x^8))/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7)).

G.f.: 3*(1 - 18*x - 76*x^2 + 788*x^3 - 1992*x^4 - 2706*x^5 + 1051*x^6 + 130*x^7 - 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - Andrew Howroyd, Mar 07 2018

Example

Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
[u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = 3.
[1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = -15;
[1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = -111.

Mathematica

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
f[n_] := FromContinuedFraction[t[n]];
t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
Coefficient[t, x, 0]; (* A267078 *)
Coefficient[t, x, 1]; (* A267079 *)
Coefficient[t, x, 2]; (* A267080 *)
Coefficient[t, x, 3]; (* A267081 *)
Coefficient[t, x, 4]; (* A267082 *)
Coefficient[t, x, 5]; (* A267083 *)
Coefficient[t, x, 6]; (* A266527 *)

Prog

(PARI) Vec(3*(1 - 18*x - 76*x^2 + 788*x^3 - 1992*x^4 - 2706*x^5 + 1051*x^6 + 130*x^7 - 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018

Crossrefs

Cf. A265762, A267078, A267079, A267080, A267081, A267082, A266527.

Sequence in context: A112936 A001063 A130168 * A089945 A135083 A323772

Adjacent sequences: A267080 A267081 A267082 * A267084 A267085 A267086

Keyword

sign,easy

Author

Clark Kimberling, Jan 11 2016

Status

approved