cp's OEIS Frontend

A266527 Coefficient of x^6 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.

%I A266527 #29 Mar 07 2018 18:29:32
%S A266527 1,5,11,131,3421,56209,1049105,18561659,334918459,5997328339,
%T A266527 107703879581,1932077585345,34673771913121,622167861459451,
%U A266527 11164539354582251,200338227165577379,3594932551574173405,64508386001097153649,1157556438367284595889
%N A266527 Coefficient of x^6 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.
%C A266527 See A265762 for a guide to related sequences.
%H A266527 Andrew Howroyd, <a href="/A266527/b266527.txt">Table of n, a(n) for n = 0..200</a>
%H A266527 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (13, 104, -260, -260, 104, 13, -1).
%F A266527 a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
%F A266527 G.f.: (1 - 8 x - 158 x^2 - 272 x^3 + 2134 x^4 + 2168 x^5 - 1009 x^6 - 130 x^7 + 10 x^8)/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7).
%F A266527 G.f.: (1 - 8*x - 158*x^2 - 272*x^3 + 2134*x^4 + 2168*x^5 - 1009*x^6 - 130*x^7 + 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - _Andrew Howroyd_, Mar 07 2018
%e A266527 Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:
%e A266527 [u,1,1,1,...] has p(0,x)  = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = 1.
%e A266527 [1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 5;
%e A266527 [1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = 11.
%t A266527 u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}];
%t A266527 f[n_] := FromContinuedFraction[t[n]];
%t A266527 t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}]
%t A266527 Coefficient[t, x, 0]; (* A267078 *)
%t A266527 Coefficient[t, x, 1]; (* A267079 *)
%t A266527 Coefficient[t, x, 2]; (* A267080 *)
%t A266527 Coefficient[t, x, 3]; (* A267081 *)
%t A266527 Coefficient[t, x, 4]; (* A267082 *)
%t A266527 Coefficient[t, x, 5]; (* A267083 *)
%t A266527 Coefficient[t, x, 6]; (* A266527 *)
%o A266527 (PARI) Vec((1 - 8*x - 158*x^2 - 272*x^3 + 2134*x^4 + 2168*x^5 - 1009*x^6 - 130*x^7 + 10*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ _Andrew Howroyd_, Mar 07 2018
%Y A266527 Cf. A265762, A267078, A267079, A267080, A267081, A267082, A267083.
%K A266527 sign,easy
%O A266527 0,2
%A A266527 _Clark Kimberling_, Jan 11 2016