A267078 Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

-5, -11, 131, 3421, 56209, 1049105, 18561659, 334918459, 5997328339, 107703879581, 1932077585345, 34673771913121, 622167861459451, 11164539354582251, 200338227165577379, 3594932551574173405, 64508386001097153649, 1157556438367284595889

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.: (-5 + 54 x + 794 x^2 + 1562 x^3 - 6048 x^4 - 5676 x^5 + 2287 x^6 + 286 x^7 - 22 x^8)/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7).

G.f.: -(5 - 54*x - 794*x^2 - 1562*x^3 + 6048*x^4 + 5676*x^5 - 2287*x^6 - 286*x^7 + 22*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) = -5.
[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) = -11;
[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) = 131.

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]; (* A267084 *)

Prog

(PARI) Vec(-(5 - 54*x - 794*x^2 - 1562*x^3 + 6048*x^4 + 5676*x^5 - 2287*x^6 - 286*x^7 + 22*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, A267079, A267080, A267081, A267082, A267083, A267084.

Sequence in context: A197538 A083418 A266527 * A276511 A020453 A036932

Adjacent sequences: A267075 A267076 A267077 * A267079 A267080 A267081

Keyword

sign,easy

Author

Clark Kimberling, Jan 10 2016

Status

approved