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 A266703 Coefficient of x^2 in minimal polynomial of the continued fraction [1^n,2/3,1,1,1,...], where 1^n means n ones. 3
 9, 11, 1, 29, 45, 149, 359, 971, 2511, 6605, 17261, 45221, 118359, 309899, 811295, 2124029, 5560749, 14558261, 38113991, 99783755, 261237231, 683927981, 1790546669, 4687712069, 12272589495, 32130056459, 84117579839, 220222683101, 576550469421, 1509428725205 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS See A265762 for a guide to related sequences. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,2,-1). FORMULA a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3). G.f.: (9 - 7 x - 39 x^2 + 14 x^3 - 4 x^4 + 2 x^5)/(1 - 2 x - 2 x^2 + x^3). a(n) = 2^(-n)*(-43*(-2)^n+(3+sqrt(5))^n*(-1+3*sqrt(5))-(3-sqrt(5))^n*(1+3*sqrt(5)))/5 for n>2. - Colin Barker, Sep 29 2016 EXAMPLE Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [2/3,1,1,1,...] = (1+3*sqrt(5))/6 has p(0,x) = -11 - 3 x + 9 x^2, so a(0) = 9; [1,2/3,1,1,...] = (19+9*sqrt(5))/22 has p(1,x) = -1 - 19 x + 11 x^2, so a(1) = 11; [1,1,2/3,1,...] = (-17+9*sqrt(5))/2 has p(2,x) = -29 + 17 x + x^2, so a(2) = 1. MATHEMATICA u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2/3}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}] Coefficient[t, x, 0] (* A266703 *) Coefficient[t, x, 1] (* A266704 *) Coefficient[t, x, 2] (* A266703 *) PROG (PARI) Vec((9-7*x-39*x^2+14*x^3-4*x^4+2*x^5)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Sep 29 2016 CROSSREFS Cf. A265762, A266704. Sequence in context: A119207 A165254 A058069 * A266701 A172283 A172185 Adjacent sequences: A266700 A266701 A266702 * A266704 A266705 A266706 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jan 09 2016 STATUS approved

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Last modified July 21 15:14 EDT 2024. Contains 374474 sequences. (Running on oeis4.)