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A266808 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

%I A266808 #8 Sep 23 2017 03:18:10
%S A266808 -168,-560,-101124,-3288624,-180132168,-7998247028,-384048485640,
%T A266808 -17892957477264,-843263161727364,-39567408316416848,
%U A266808 -1859687400468342888,-87350263553726629620,-4103880417768964672104,-192790045902230868971504,-9057117701582885083841028
%N A266808 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.
%C A266808 See A265762 for a guide to related sequences.
%H A266808 G. C. Greubel, <a href="/A266808/b266808.txt">Table of n, a(n) for n = 0..595</a>
%H A266808 <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 A266808 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 A266808 G.f.:  -((4 (-42 + 1288 x + 9467 x^2 - 57564 x^3 - 198636 x^4 + 39086 x^5 - 5774 x^6 - 48 x^7 + 3 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 2376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).
%e A266808 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 A266808 [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) = -168.
%e A266808 [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) = -560;
%e A266808 [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) = -101124.
%t A266808 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
%t A266808 f[n_] := FromContinuedFraction[t[n]];
%t A266808 t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
%t A266808 Coefficient[t, x, 0];  (* A266803 *)
%t A266808 Coefficient[t, x, 1];  (* A266808 *)
%t A266808 Coefficient[t, x, 2];  (* A267061 *)
%t A266808 Coefficient[t, x, 3];  (* A267062 *)
%t A266808 Coefficient[t, x, 4];  (* A267063 *)
%t A266808 Coefficient[t, x, 5];  (* A267064 *)
%t A266808 Coefficient[t, x, 6];  (* A267065 *)
%t A266808 Coefficient[t, x, 7];  (* A267066 *)
%t A266808 Coefficient[t, x, 8];  (* A266803 *)
%Y A266808 Cf. A265762, A266803, A267061, A267062, A267063, A267064, A267065, A267066.
%K A266808 sign,easy
%O A266808 0,1
%A A266808 _Clark Kimberling_, Jan 10 2016