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Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,5,1,1,1,...], where 1^n means n ones.
3

%I #11 Sep 08 2022 08:46:15

%S 1,19,29,95,229,619,1601,4211,11005,28831,75461,197579,517249,1354195,

%T 3545309,9281759,24299941,63618091,166554305,436044851,1141580221,

%U 2988695839,7824507269,20484825995,53629970689,140405086099,367585287581,962350776671

%N Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,5,1,1,1,...], where 1^n means n ones.

%C See A265762 for a guide to related sequences.

%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).

%F G.f.: (1 + 17 x - 11 x^2)/(1 - 2 x - 2 x^2 + x^3).

%e Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:

%e [5,1,1,1,1,...] = (9+sqrt(5))/2 has p(0,x) = 19 - 9 x + x^2, so a(0) = 1;

%e [1,5,1,1,1,...] = (47-sqrt(5))/38 has p(1,x) = 29 - 47 x + 19 x^2, so a(1) = 19;

%e [1,1,5,1,1,...] = (105+sqrt(5))/58 has p(2,x) = 5 - 105 x + 29 x^2, so a(2) = 29.

%t u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {5}, {{1}}];

%t f[n_] := FromContinuedFraction[t[n]];

%t t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]

%t Coefficient[t, x, 0] (* A265804 *)

%t Coefficient[t, x, 1] (* A265805 *)

%t Coefficient[t, x, 2] (* A236804 *)

%t LinearRecurrence[{2, 2, -1}, {1, 19, 29}, 30] (* _Vincenzo Librandi_, Jan 06 2016 *)

%o (Magma) I:=[1,19,29]; [n le 3 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 (PARI) Vec((1+17*x-11*x^2)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ _Altug Alkan_, Jan 07 2016

%Y Cf. A265762, A265805.

%K nonn

%O 0,2

%A _Clark Kimberling_, Jan 05 2016