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%I #35 Sep 08 2022 08:46:15
%S 1,11,19,59,145,389,1009,2651,6931,18155,47521,124421,325729,852779,
%T 2232595,5845019,15302449,40062341,104884561,274591355,718889491,
%U 1882077131,4927341889,12899948549,33772503745,88417562699,231480184339,606022990331,1586588786641
%N Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,4,1,1,1,...], where 1^n means n ones.
%C See A265762 for a guide to related sequences.
%H Colin Barker, <a href="/A265802/b265802.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) for n>3.
%F G.f.: (1 + 9*x - 5*x^2)/(1 - 2*x - 2*x^2 + x^3).
%F a(n) = (2^(-n)*(-13*(-2)^n + 3*(3-sqrt(5))^(1+n) + 3*(3+sqrt(5))^(1+n)))/5. - _Colin Barker_, Oct 20 2016
%F From _Klaus Purath_, Oct 28 2019: (Start)
%F (a(n-3) - a(n-2) - a(n-1) + a(n))/6 = Fibonacci(2*n-1).
%F (a(n-5) + a(n))/30 = Fibonacci(2*n-3).
%F (a(n) - a(n-4))/18 = Fibonacci(2*n-2). (End)
%F E.g.f.: (1/5)*exp(-x)*(-13 + exp(-(1/2)*(-5 + sqrt(5))*x)*(9 - 3*sqrt(5) + 3*(3 + sqrt(5))*exp(sqrt(5)*x))). - _Stefano Spezia_, Dec 09 2019
%F a(n) = 6*Fibonacci(n+1)^2 - 5*(-1)^n = (6*Lucas(2*n+2) - 13*(-1)^n)/5. - _G. C. Greubel_, Dec 11 2019
%e Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
%e [4,1,1,1,1,...] = (7 + sqrt(5))/2 has p(0,x) = 11 - 7 x + x^2, so a(0) = 1;
%e [1,4,1,1,1,...] = (29 - sqrt(5))/22 has p(1,x) = 19 - 29 x + 11 x^2, so a(1) = 11;
%e [1,1,4,1,1,...] = (67 + sqrt(5))/38 has p(2,x) = 59 - 67 x + 19 x^2, so a(2) = 19.
%p with(combinat); seq(6*fibonacci(n+1)^2 - 5*(-1)^n, n=0..30); # _G. C. Greubel_, Dec 11 2019
%t u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {4}, {{1}}];
%t f[n_] := FromContinuedFraction[t[n]];
%t t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
%t Coefficient[t, x, 0] (* A265802 *)
%t Coefficient[t, x, 1] (* A265803 *)
%t Coefficient[t, x, 2] (* A236802 *)
%t Join[{1}, LinearRecurrence[{2, 2, -1}, {11, 19, 59}, 30]] (* _Vincenzo Librandi_, Jan 06 2016 *)
%t Table[6*Fibonacci[n+1]^2 - 5*(-1)^n, {n,0,30}] (* _G. C. Greubel_, Dec 11 2019 *)
%o (PARI) Vec((1+9*x-5*x^2)/(1-2*x-2*x^2+x^3) + O(x^30)) \\ _Altug Alkan_, Jan 04 2016
%o (PARI) vector(31, n, 6*fibonacci(n)^2 + 5*(-1)^n) \\ _G. C. Greubel_, Dec 11 2019
%o (Magma) I:=[1,11,19,59]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jan 06 2016
%o (Sage) [6*fibonacci(n+1)^2 - 5*(-1)^n for n in (0..30)] # _G. C. Greubel_, Dec 11 2019
%o (GAP) List([0..30], n-> 6*Fibonacci(n+1)^2 - 5*(-1)^n); # _G. C. Greubel_, Dec 11 2019
%Y Cf. A000032, A000045, A265762, A265803.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Jan 04 2016
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