A267062 - OEIS

%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