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A267082 Coefficient of x^4 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

%I A267082 #10 Mar 07 2018 18:45:03
%S A267082 0,6,456,6240,131238,2238780,41011296,730283034,13143304440,
%T A267082 235581102912,4229156006790,75876624195564,1361636473680576,
%U A267082 24432987781993530,438436202143461288,7867390833380267040,141174789462751501926,2533277512666920359964
%N A267082 Coefficient of x^4 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.
%C A267082 See A265762 for a guide to related sequences.
%H A267082 Andrew Howroyd, <a href="/A267082/b267082.txt">Table of n, a(n) for n = 0..200</a>
%H A267082 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (14, 90, -350, 90, 14, -1).
%F A267082 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 A267082 G.f.:  (6 (x + 62 x^2 - 114 x^3 + 823 x^4 - 182 x^5 - 28 x^6 + 2 x^7))/(1 - 14 x - 90 x^2 + 350 x^3 - 90 x^4 - 14 x^5 + x^6).
%F A267082 From _Andrew Howroyd_, Mar 07 2018: (Start)
%F A267082 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 A267082 G.f.: 6*x*(1 + 62*x - 114*x^2 + 823*x^3 - 182*x^4 - 28*x^5 + 2*x^6)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)).
%F A267082 (End)
%e A267082 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 A267082 [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) = 0.
%e A267082 [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) = 6;
%e A267082 [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) = 456.
%t A267082 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
%t A267082 f[n_] := FromContinuedFraction[t[n]];
%t A267082 t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
%t A267082 Coefficient[t, x, 0]; (* A267078 *)
%t A267082 Coefficient[t, x, 1]; (* A267079 *)
%t A267082 Coefficient[t, x, 2]; (* A267080 *)
%t A267082 Coefficient[t, x, 3]; (* A267081 *)
%t A267082 Coefficient[t, x, 4]; (* A267082 *)
%t A267082 Coefficient[t, x, 5]; (* A267083 *)
%t A267082 Coefficient[t, x, 6]; (* A266527 *)
%o A267082 (PARI) concat([0], 6*Vec((1 + 62*x - 114*x^2 + 823*x^3 - 182*x^4 - 28*x^5 + 2*x^6)/((1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30))) \\ _Andrew Howroyd_, Mar 07 2018
%Y A267082 Cf. A265762, A267078, A267079, A267080, A267081, A267083, A266527.
%K A267082 nonn,easy
%O A267082 0,2
%A A267082 _Clark Kimberling_, Jan 11 2016