A266799 - OEIS

%I #7 Oct 21 2019 15:22:59

%S 1,1,49,229,1861,12001,84241,572209,3935569,26939221,184737301,

%T 1265964481,8677687969,59476087009,407659540081,2794128600901,

%U 19151272325221,131264694791329,899701808208049,6166647394567441,42266831441062801,289701168799073461

%N Coefficient of x^0 in the minimal polynomial of the continued fraction [1^n,sqrt(3),1,1,...], where 1^n means n ones.

%C See A265762 for a guide to related sequences.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).

%F a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) .

%F G.f.:  (-1 + 4 x - 29 x^2 + 16 x^3 - x^4)/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5).

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

%e [sqrt(3),1,1,1,...] has p(0,x)=1-8x-7x^2+2x^3+x^4, so a(0) = 1;

%e [1,sqrt(3),1,1,1,...] has p(1,x)=1-12x+23x^2-12x^3+x^4, so a(1) = 1;

%e [1,1,sqrt(3),1,1,1...] has p(2,x)=49-98x+65x^2-16x^3+x^4, so a(2) = 49.

%t u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[3]}, {{1}}];

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

%t t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];

%t Coefficient[t, x, 0] ; (* A266799 *)

%t Coefficient[t, x, 1];  (* A266800 *)

%t Coefficient[t, x, 2];  (* A266801 *)

%t Coefficient[t, x, 3];  (* A266802 *)

%t Coefficient[t, x, 4];  (* A266799 *)

%t LinearRecurrence[{5,15,-15,-5,1},{1,1,49,229,1861},30] (* _Harvey P. Dale_, Oct 21 2019 *)

%Y Cf. A265762, A266800, A266801, A266802.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Jan 09 2016