A267080 - OEIS

%I #10 Mar 07 2018 18:43:26

%S -6,-66,1110,18318,333750,5938446,106865274,1915425570,34385669382,

%T 616923941070,11070947149014,198655308975486,3564757609030650,

%U 63966755470710018,1147838391054195510,20597113658105850126,369600280281802257654,6632207432249371045230

%N Coefficient of x^2 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="/A267080/b267080.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (14, 90, -350, 90, 14, -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.:  -((6 (1 - 3 x - 429 x^2 - 1103 x^3 + 7527 x^4 - 1975 x^5 - 308 x^6 + 22 x^7))/(1 - 14 x - 90 x^2 + 350 x^3 - 90 x^4 - 14 x^5 + x^6)).

%F From _Andrew Howroyd_, Mar 07 2018: (Start)

%F a(n) = 14*a(n-1) + 90*a(n-2) - 350*a(n-3) + 90*a(n-4) + 14*a(n-5) - a(n-6) for n > 7.

%F G.f.: -6*(1 - 3*x - 429*x^2 - 1103*x^3 + 7527*x^4 - 1975*x^5 - 308*x^6 + 22*x^7)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)).

%F (End)

%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) = -6.

%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) = -66;

%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) = 1110.

%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(-6*(1 - 3*x - 429*x^2 - 1103*x^3 + 7527*x^4 - 1975*x^5 - 308*x^6 + 22*x^7)/((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, A267079, A267081, A267082, A267083, A266527.

%K sign,easy

%O 0,1

%A _Clark Kimberling_, Jan 11 2016