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%I #8 Sep 23 2017 03:18:24
%S 212,-5760,-165852,-10501476,-449827456,-21948311748,-1016699956620,
%T -48023357086272,-2251419462422716,-105852417560435076,
%U -4971310326775823808,-233572686675369390180,-10972461323000994899692,-515480788238950647507456,-24216468853316695676874396
%N Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.
%C See A265762 for a guide to related sequences.
%H G. C. Greubel, <a href="/A267062/b267062.txt">Table of n, a(n) for n = 0..595</a>
%H <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 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 G.f.: (4 (-53 + 3242 x + 30345 x^2 - 58506 x^3 - 383152 x^4 - 61754 x^5 + 46551 x^6 - 1122 x^7 + 9 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 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 [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) = 212.
%e [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) = -5760;
%e [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) = -165852.
%t u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];
%t f[n_] := FromContinuedFraction[t[n]];
%t t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
%t Coefficient[t, x, 0]; (* A266803 *)
%t Coefficient[t, x, 1]; (* A266808 *)
%t Coefficient[t, x, 2]; (* A267061 *)
%t Coefficient[t, x, 3]; (* A267062 *)
%t Coefficient[t, x, 4]; (* A267063 *)
%t Coefficient[t, x, 5]; (* A267064 *)
%t Coefficient[t, x, 6]; (* A267065 *)
%t Coefficient[t, x, 7]; (* A267066 *)
%t Coefficient[t, x, 8]; (* A266803 *)
%Y Cf. A265762, A266803, A266808, A267061, A267063, A267064, A267065, A267066.
%K sign,easy
%O 0,1
%A _Clark Kimberling_, Jan 10 2016
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