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%I #10 Mar 07 2018 18:44:41
%S -9,35,-969,-14359,-279261,-4862231,-88270665,-1576950691,
%T -28345226121,-508305487319,-9123426587229,-163697793422935,
%U -2937543639603849,-52711355807057699,-945871877489577801,-16972948054702729111,-304567428780675699165,-5465239154667149397911
%N Coefficient of x^3 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="/A267081/b267081.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10, 135, 135, 10, -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.: (-9 + 125 x - 104 x^2 - 8179 x^3 - 9491 x^4 - 700 x^5 + 70 x^6)/(1 - 10 x - 135 x^2 - 135 x^3 - 10 x^4 + x^5).
%F From _Andrew Howroyd_, Mar 07 2018: (Start)
%F a(n) = 10*a(n-1) + 135*a(n-2) + 135*a(n-3) + 10*a(n-4) - a(n-5) for n > 6.
%F G.f.: (-9 + 125*x - 104*x^2 - 8179*x^3 - 9491*x^4 - 700*x^5 + 70*x^6)/((1 + x)*(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) = -9.
%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) = 35;
%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) = -969.
%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((-9 + 125*x - 104*x^2 - 8179*x^3 - 9491*x^4 - 700*x^5 + 70*x^6)/((1 + x)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ _Andrew Howroyd_, Mar 07 2018
%Y Cf. A265762, A267078, A267079, A267080, A267082, A267083, A266527.
%K sign,easy
%O 0,1
%A _Clark Kimberling_, Jan 11 2016
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