A267083 - OEIS

%I #9 Mar 07 2018 18:45:42

%S 3,-15,-111,-1419,-32847,-549633,-10161885,-180368529,-3250331151,

%T -58231071723,-1045558051887,-18757368165345,-336617548680381,

%U -6040149618970929,-108387507398297007,-1944925169961946443,-34900332815651650575,-626260604480315081409

%N 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.

%C See A265762 for a guide to related sequences.

%H Andrew Howroyd, <a href="/A267083/b267083.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 (-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)).

%F 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

%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) = 3.

%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) = -15;

%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) = -111.

%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*(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

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

%K sign,easy

%O 0,1

%A _Clark Kimberling_, Jan 11 2016