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A266708 Coefficient of x in minimal polynomial of the continued fraction [1^n,tau,1,1,1,...], where 1^n means n ones and tau = golden ratio = (1 + sqrt(5))/2.

%I A266708 #18 Nov 16 2024 20:13:37
%S A266708 0,-10,-18,-56,-138,-370,-960,-2522,-6594,-17272,-45210,-118370,
%T A266708 -309888,-811306,-2124018,-5560760,-14558250,-38114002,-99783744,
%U A266708 -261237242,-683927970,-1790546680,-4687712058,-12272589506,-32130056448,-84117579850,-220222683090
%N A266708 Coefficient of x in minimal polynomial of the continued fraction [1^n,tau,1,1,1,...], where 1^n means n ones and tau = golden ratio = (1 + sqrt(5))/2.
%C A266708 See A265762 for a guide to related sequences.
%H A266708 Colin Barker, <a href="/A266708/b266708.txt">Table of n, a(n) for n = 0..1000</a>
%H A266708 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A266708 G.f.: 2*x*(-5 + x)/((1 + x)*(1 - 3*x + x^2)).
%F A266708 a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3).
%F A266708 a(n) = -2*A192914(n+1).
%F A266708 a(n) = (2^(1-n)*(3*(-1)^n*2^(1+n)+(3-sqrt(5))^n*(-3+2*sqrt(5))-(3+sqrt(5))^n*(3+2*sqrt(5))))/5. - _Colin Barker_, Sep 30 2016
%e A266708 Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
%e A266708 [tau,1,1,1,1,...] = sqrt(5) has p(0,x) = -5 + x^2, so a(0) = 0;
%e A266708 [1,tau,1,1,1,...] = (5 + sqrt(5))/5 has p(1,x) = 4 - 10*x + 5*x^2, so a(1) = -10;
%e A266708 [1,1,tau,1,1,...] = (9 - sqrt(5))/4 has p(2,x) = 19 - 18*x + 4*x^2, so a(2) = -18.
%t A266708 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {GoldenRatio}, {{1}}];
%t A266708 f[n_] := FromContinuedFraction[t[n]];
%t A266708 t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
%t A266708 Coefficient[t, x, 0] (* A266707 *)
%t A266708 Coefficient[t, x, 1] (* A266708 *)
%t A266708 Coefficient[t, x, 2] (* A266707 *)
%o A266708 (PARI) a(n) = round((2^(1-n)*(3*(-1)^n*2^(1+n)+(3-sqrt(5))^n*(-3+2*sqrt(5))-(3+sqrt(5))^n*(3+2*sqrt(5))))/5) \\ _Colin Barker_, Sep 30 2016
%o A266708 (PARI) concat(0, Vec(-2*x*(5-x)/((1+x)*(1-3*x+x^2)) + O(x^30))) \\ _Colin Barker_, Sep 30 2016
%Y A266708 Cf. A192914, A265762, A266708.
%K A266708 sign,easy
%O A266708 0,2
%A A266708 _Clark Kimberling_, Jan 09 2016