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A265762 Coefficient of x in minimal polynomial of the continued fraction [1^n,2,1,1,1,...], where 1^n means n ones. 43
-3, -5, -15, -37, -99, -257, -675, -1765, -4623, -12101, -31683, -82945, -217155, -568517, -1488399, -3896677, -10201635, -26708225, -69923043, -183060901, -479259663, -1254718085, -3284894595, -8599965697, -22515002499, -58945041797, -154320122895 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

In the following guide to related sequences, d(n), e(n), f(n) represent the coefficients in the minimal polynomial written as d(n)*x^2 + e(n)*x + f(n), except, in some cases, for initial terms. All of these sequences (eventually) satisfy the linear recurrence relation a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).

continued fractions       d(n)     e(n)     f(n)

[1^n,2,1,1,1,...]       A236428  A265762  A236428

[1^n,3,1,1,1,...]       A236428  A265762  A236428

[1^n,4,1,1,1,...]       A265802  A265803  A265802

[1^n,5,1,1,1,...]       A265804  A265805  A265804

[1^n,1/2,1,1,1,...]     A266699  A266700  A266699

[1^n,1/3,1,1,1,...]     A266701  A266702  A266701

[1^n,2/3,1,1,1,...]     A266703  A266704  A266703

[1^n,sqrt(5),1,1,1,...] A266705  A266706  A266705

[1^n,tau,1,1,1,...]     A266707  A266708  A266707

[2,1^n,2,1,1,1,...]     A236428  A266709  A236428

The following forms of continued fractions have minimal polynomials of degree 4 and, after initial terms, satisfy the following linear recurrence relation:

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).

[1^n,sqrt(2),1,1,1,...]:  A266710, A266711, A266712, A266713, A266710

[1^n,sqrt(3),1,1,1,...]:  A266799, A266800, A266801, A266802, A266799

[1^n,sqrt(6),1,1,1,...]:  A266804, A266805, A266806, A266807, A277804

Continued fractions [1^n,2^(1/3),1,1,1,...] have minimal polynomials of degree 6.  The coefficient sequences are linearly recurrenct with signature {13, 104, -260, -260, 104, 13, -1, 0, 0}; see A267078-A267083.

Continued fractions [1^n,sqrt(2)+sqrt(3),1,1,1,...] have minimal polynomials of degree 8.  The coefficient sequences are linearly recurrenct with signature {13, 104, -260, -260, 104, 13, -1}; see A266803, A266808, A267061-A267066.

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.:  (-3 + x + x^2)/(1 - 2 x - 2 x^2 + x^3).

a(n) = (-1)*(2^(-n)*(3*(-2)^n+2*((3-sqrt(5))^(1+n)+(3+sqrt(5))^(1+n))))/5. - Colin Barker, Sep 27 2016

EXAMPLE

Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:

[2,1,1,1,1,...] = (3 + sqrt(5))/2 has p(0,x) = x^2 - 3x + 1, so a(0) = -3;

[1,2,1,1,1,...] = (5 - sqrt(5))/2 has p(1,x) = x^2 - 5x + 5, so a(1) = -5;

[1,1,2,1,1,...] = (15 + sqrt(5))/10 has p(2,x) = 5x^2 - 15x + 11, so a(2) = -15.

MATHEMATICA

Program 1:

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2}, {{1}}];

f[n_] := FromContinuedFraction[t[n]];

t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}]

Coefficient[t, x, 0] (* A236428 *)

Coefficient[t, x, 1] (* A265762 *)

Coefficient[t, x, 2] (* A236428 *)

Program 2:

LinearRecurrence[{2, 2, -1}, {-3, -5, -15}, 50] (* Vincenzo Librandi, Jan 05 2016 *)

PROG

(PARI) Vec((-3+x+x^2)/(1-2*x-2*x^2+x^3) + O(x^100)) \\ Altug Alkan, Jan 04 2016

(MAGMA) I:=[-3, -5, -15]; [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 05 2016

CROSSREFS

Cf. A236428, A265802.

Sequence in context: A089485 A279684 A146212 * A018516 A138017 A280764

Adjacent sequences:  A265759 A265760 A265761 * A265763 A265764 A265765

KEYWORD

sign,easy

AUTHOR

Clark Kimberling, Jan 04 2016

STATUS

approved

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Last modified August 18 16:13 EDT 2017. Contains 290727 sequences.