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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.)