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 A267079 Coefficient of x in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones. 7
 -15, 45, -633, -12321, -212379, -3867255, -68998575, -1240820397, -22247101689, -399334774401, -7164902653275, -128574917201655, -2307142450214223, -41400271270803501, -742895806968482169, -13330737506206610145, -239210288473732159515 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS See A265762 for a guide to related sequences. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..200 Index entries for linear recurrences with constant coefficients, signature (13, 104, -260, -260, 104, 13, -1). FORMULA 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. G.f.: (3 (-5 + 80 x + 114 x^2 - 4224 x^3 + 7142 x^4 + 7912 x^5 - 3123 x^6 - 390 x^7 + 30 x^8))/(1 - 13 x - 104 x^2 + 260 x^3 + 260 x^4 - 104 x^5 - 13 x^6 + x^7). G.f.: -3*(5 - 80*x - 114*x^2 + 4224*x^3 - 7142*x^4 - 7912*x^5 + 3123*x^6 + 390*x^7 - 30*x^8)/((1 + x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)). - Andrew Howroyd, Mar 07 2018 EXAMPLE Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [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) = -15. [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) = 45; [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) = -633. MATHEMATICA u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}] Coefficient[t, x, 0]; (* A267078 *) Coefficient[t, x, 1]; (* A267079 *) Coefficient[t, x, 2]; (* A267080 *) Coefficient[t, x, 3]; (* A267081 *) Coefficient[t, x, 4]; (* A267082 *) Coefficient[t, x, 5]; (* A267083 *) Coefficient[t, x, 6]; (* A266527 *) PROG (PARI) Vec(-3*(5 - 80*x - 114*x^2 + 4224*x^3 - 7142*x^4 - 7912*x^5 + 3123*x^6 + 390*x^7 - 30*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 CROSSREFS Cf. A265762, A267078, A267080, A267081, A267082, A267083, A266527. Sequence in context: A095129 A219813 A068513 * A290583 A033480 A041434 Adjacent sequences: A267076 A267077 A267078 * A267080 A267081 A267082 KEYWORD sign,easy AUTHOR Clark Kimberling, Jan 11 2016 STATUS approved

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Last modified August 10 22:52 EDT 2024. Contains 375059 sequences. (Running on oeis4.)