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A266706 Coefficient of x in minimal polynomial of the continued fraction [1^n,sqrt(5),1,1,1,...], where 1^n means n ones.

%I A266706 #18 May 21 2020 16:52:23
%S A266706 1,-23,-45,-135,-337,-899,-2337,-6135,-16045,-42023,-110001,-288003,
%T A266706 -753985,-1973975,-5167917,-13529799,-35421457,-92734595,-242782305,
%U A266706 -635612343,-1664054701,-4356551783,-11405600625,-29860250115,-78175149697,-204665198999
%N A266706 Coefficient of x in minimal polynomial of the continued fraction [1^n,sqrt(5),1,1,1,...], where 1^n means n ones.
%C A266706 See A265762 for a guide to related sequences.
%H A266706 Colin Barker, <a href="/A266706/b266706.txt">Table of n, a(n) for n = 1..1000</a>
%H A266706 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F A266706 a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3).
%F A266706 G.f.: x*(1 - 25*x - x^2 + 2*x^3) / ((1 + x)*(1 - 3*x + x^2)).
%F A266706 a(n) = (1/5)*2^(-n)*(-23*(-2)^n + (4-8*sqrt(5)) * (3+sqrt(5))^n + (3-sqrt(5))^n*(4+8*sqrt(5))) for n>1. - _Colin Barker_, May 21 2020
%F A266706 a(n) = 3*a(n-1) - a(n-2) - 23*(-1)^n for n >= 4. - _Greg Dresden_, May 18 2020
%e A266706 Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
%e A266706 [sqrt(5),1,1,1,1,...] = (-1+3*sqrt(5))/2 has p(0,x)=-11+x+x^2, so a(0) = 1;
%e A266706 [1,sqrt(5),1,1,1,...] = (23+3*sqrt(5))/22 has p(1,x)=11-23x+11x^2, so a(1) = 11;
%e A266706 [1,1,sqrt(5),1,1,...] = (45-3* sqrt(5))/22 has p(2,x)=45-45x+11x^2, so a(2) = 11.
%t A266706 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[5]}, {{1}}];
%t A266706 f[n_] := FromContinuedFraction[t[n]];
%t A266706 t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]
%t A266706 Coefficient[t, x, 0] (* A266705 *)
%t A266706 Coefficient[t, x, 1] (* A266706 *)
%t A266706 Coefficient[t, x, 2] (* A266705 *)
%t A266706 LinearRecurrence[{2,2,-1},{1,-23,-45,-135},40] (* _Harvey P. Dale_, Jul 30 2017 *)
%o A266706 (PARI) Vec(x*(1-25*x-x^2+2*x^3)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Sep 29 2016
%Y A266706 Cf. A265762, A266705.
%K A266706 sign,easy
%O A266706 1,2
%A A266706 _Clark Kimberling_, Jan 09 2016