%I #8 Mar 07 2018 18:41:34

%S -15,45,-633,-12321,-212379,-3867255,-68998575,-1240820397,

%T -22247101689,-399334774401,-7164902653275,-128574917201655,

%U -2307142450214223,-41400271270803501,-742895806968482169,-13330737506206610145,-239210288473732159515

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

%C See A265762 for a guide to related sequences.

%H Andrew Howroyd, <a href="/A267079/b267079.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (13, 104, -260, -260, 104, 13, -1).

%F 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.

%F G.f.: (3 (-5 + 80 x + 114 x^2 - 4224 x^3 + 7142 x^4 + 7912 x^5 - 3123 x^6 - 390 x^7 + 30 x^8))/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7).

%F G.f.: -3*(5 - 80*x - 114*x^2 + 4224*x^3 - 7142*x^4 - 7912*x^5 + 3123*x^6 + 390*x^7 - 30*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - _Andrew Howroyd_, Mar 07 2018

%e Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:

%e [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) = -15.

%e [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) = 45;

%e [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) = -633.

%t u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];

%t f[n_] := FromContinuedFraction[t[n]];

%t t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]

%t Coefficient[t, x, 0]; (* A267078 *)

%t Coefficient[t, x, 1]; (* A267079 *)

%t Coefficient[t, x, 2]; (* A267080 *)

%t Coefficient[t, x, 3]; (* A267081 *)

%t Coefficient[t, x, 4]; (* A267082 *)

%t Coefficient[t, x, 5]; (* A267083 *)

%t Coefficient[t, x, 6]; (* A266527 *)

%o (PARI) Vec(-3*(5 - 80*x - 114*x^2 + 4224*x^3 - 7142*x^4 - 7912*x^5 + 3123*x^6 + 390*x^7 - 30*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

%Y Cf. A265762, A267078, A267080, A267081, A267082, A267083, A266527.

%K sign,easy

%O 0,1

%A _Clark Kimberling_, Jan 11 2016