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 A265803 Coefficient of x in minimal polynomial of the continued fraction [1^n,4,1,1,1,...], where 1^n means n ones. 3
 -7, -29, -67, -185, -475, -1253, -3271, -8573, -22435, -58745, -153787, -402629, -1054087, -2759645, -7224835, -18914873, -49519771, -129644453, -339413575, -888596285, -2326375267, -6090529529, -15945213307, -41745110405, -109290117895, -286125243293 (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.:  (-7 - 15 x + 5 x^2)/(1 - 2 x - 2 x^2 + x^3). a(n) = (2^(-n)*(13*(-2)^n + 12*(3-sqrt(5))^n*(-2+sqrt(5)) - 12*(2+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Oct 20 2016 EXAMPLE Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [4,1,1,1,1,...] = (7 + sqrt(5))/2 has p(0,x) = 11 - 7 x + x^2, so a(0) = 1; [1,4,1,1,1,...] = (29 - sqrt(5))/22 has p(1,x) = 19 - 29 x + 11 x^2, so a(1) = 11; [1,1,4,1,1,...] = (67 + sqrt(5))/38 has p(2,x) = 59 - 67 x + 19 x^2, so a(2) = 19. MATHEMATICA u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {4}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}] Coefficient[t, x, 0] (* A265802 *) Coefficient[t, x, 1] (* A265803 *) Coefficient[t, x, 2] (* A236802 *) LinearRecurrence[{2, 2, -1}, {-7, -29, -67}, 30] (* Vincenzo Librandi, Jan 06 2016 *) PROG (PARI) Vec((-7-15*x+5*x^2)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ Altug Alkan, Jan 04 2016 (MAGMA) I:=[-7, -29, -67]; [n le 3 select I[n] else 2*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 06 2016 CROSSREFS Cf. A265762, A265802. Sequence in context: A041621 A022272 A185438 * A176616 A231988 A141854 Adjacent sequences:  A265800 A265801 A265802 * A265804 A265805 A265806 KEYWORD sign,easy AUTHOR Clark Kimberling, Jan 04 2016 STATUS approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)