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A267063 Coefficient of x^4 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

%I A267063 #7 Sep 22 2017 22:22:31
%S A267063 47,7547,96847,8834047,335645147,17176306847,781541264047,
%T A267063 37170460359547,1738056704580047,81798124546203647,
%U A267063 3840142385820445147,180452111090491814047,8476561791232835731247,398233155957829357831547,18708208945112842389197647
%N A267063 Coefficient of x^4 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.
%C A267063 See A265762 for a guide to related sequences.
%H A267063 G. C. Greubel, <a href="/A267063/b267063.txt">Table of n, a(n) for n = 0..595</a>
%H A267063 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (34, 714, -4641, -12376, 12376, 4641, -714, -34, 1).
%F A267063 a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
%F A267063 G.f.:  (-47 - 5949 x + 193309 x^2 - 370818 x^3 - 1746090 x^4 + 850782 x^5 + 32909 x^6 - 15549 x^7 + 353 x^8)/(-1 + 34 x + 714 x^2 - 4641 x^3 - 12376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9).
%e A267063 Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
%e A267063 [u+v,1,1,1,...] has p(0,x)  = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = 47.
%e A267063 [1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = 7547;
%e A267063 [1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = 96847.
%t A267063 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
%t A267063 f[n_] := FromContinuedFraction[t[n]];
%t A267063 t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
%t A267063 Coefficient[t, x, 0];  (* A266803 *)
%t A267063 Coefficient[t, x, 1];  (* A266808 *)
%t A267063 Coefficient[t, x, 2];  (* A267061 *)
%t A267063 Coefficient[t, x, 3];  (* A267062 *)
%t A267063 Coefficient[t, x, 4];  (* A267063 *)
%t A267063 Coefficient[t, x, 5];  (* A267064 *)
%t A267063 Coefficient[t, x, 6];  (* A267065 *)
%t A267063 Coefficient[t, x, 7];  (* A267066 *)
%t A267063 Coefficient[t, x, 8];  (* A266803 *)
%Y A267063 Cf. A265762, A266803, A266808, A267061, A267062, A267064, A267065, A267066.
%K A267063 nonn,easy
%O A267063 0,1
%A A267063 _Clark Kimberling_, Jan 10 2016