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A266711 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.

%I A266711 #11 Sep 08 2022 08:46:15
%S A266711 -6,2,18,-102,-714,-4826,-33222,-227298,-1558962,-10682534,-73226346,
%T A266711 -501882042,-3439999878,-23577981122,-161606223954,-1107664654566,
%U A266711 -7592048797962,-52036670543258,-356664661728582,-2444615917773474,-16755646877311986,-114844911923314982
%N A266711 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2),1,1,...], where 1^n means n ones.
%C A266711 See A265762 for a guide to related sequences.
%H A266711 G. C. Greubel, <a href="/A266711/b266711.txt">Table of n, a(n) for n = 0..1000</a>
%H A266711 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).
%F A266711 a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).
%F A266711 G.f.:  2*(3 -16*x -49*x^2 +156*x^3 +237*x^4 -280*x^5 -88*x^6 +18*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5).
%e A266711 Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
%e A266711 [sqrt(2),1,1,1,...] has p(0,x) = -1 - 6 x - 5 x^2 + 2 x^3 + x^4, so a(0) = -6;
%e A266711 [1,sqrt(2),1,1,1,...] has p(1,x) = 1 + 2 x - 7 x^2 + 2 x^3 + x^4, so a(1) = 2;
%e A266711 [1,1,sqrt(2),1,1,1...] has p(2,x) = -9 + 18 x - 7 x^2 - 2 x^3 + x^4, so a(2) = 18.
%t A266711 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2]}, {{1}}];
%t A266711 f[n_] := FromContinuedFraction[t[n]];
%t A266711 t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];
%t A266711 Coefficient[t, x, 0] ; (* A266710 *)
%t A266711 Coefficient[t, x, 1];  (* A266711 *)
%t A266711 Coefficient[t, x, 2];  (* A266712 *)
%t A266711 Coefficient[t, x, 3];  (* A266713 *)
%t A266711 Coefficient[t, x, 4];  (* A266710 *)
%t A266711 LinearRecurrence[{5,15,-15,-5,1}, {-6, 2, 18, -102, -714, -4826, -33222, -227298}, 30] (* _G. C. Greubel_, Jan 26 2018 *)
%o A266711 (PARI) x='x+O('x^30); Vec(2*(3 -16*x -49*x^2 +156*x^3 +237*x^4 -280*x^5 -88*x^6 +18*x^7)/(-1 +5*x +15*x^2 -15*x^3 -5*x^4 +x^5)) \\ _G. C. Greubel_, Jan 26 2018
%o A266711 (Magma) I:=[-102, -714, -4826, -33222, -227298]; [-6, 2, 18] cat [n le 5 select I[n] else 5*Self(n-1) + 15*Self(n-2) - 15*Self(n-3) - 5*Self(n-4) + Self(n-5): n in [1..30]]; // _G. C. Greubel_, Jan 26 2018
%Y A266711 Cf. A265762, A266710, A266712, A266713.
%K A266711 sign,easy
%O A266711 0,1
%A A266711 _Clark Kimberling_, Jan 09 2016