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A266700 Coefficient of x in minimal polynomial of the continued fraction [1^n,1/2,1,1,1,...], where 1^n means n ones. 3
0, -10, -12, -44, -102, -280, -720, -1898, -4956, -12988, -33990, -89000, -232992, -609994, -1596972, -4180940, -10945830, -28656568, -75023856, -196415018, -514221180, -1346248540, -3524524422, -9227324744, -24157449792, -63245024650, -165577624140 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.: (2 (-5 x + 4 x^2))/(1 - 2 x - 2 x^2 + x^3).

a(n) = (2^(-n)*(9*(-1)^n*2^(1+n) + (3-sqrt(5))^n*(-9+sqrt(5)) - (3+sqrt(5))^n*(9+sqrt(5))))/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:

[1/2,1,1,1,1,...] = sqrt(5))/2 has p(0,x) = -5 + 4 x^2, so a(0) = 1;

[1,1/2,1,1,1,...] = (5 + 2 sqrt(5))/5 has p(1,x) = 1 - 10 x + 5 x^2, so a(1) = 19;

[1,1,1/2,1,1,...] = 6 - 2 sqrt(5) has p(2,x) = 16 - 12 x + x^2, so a(2) = 29.

MATHEMATICA

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

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

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

Coefficient[t, x, 0] (* A266699 *)

Coefficient[t, x, 1] (* A266700 *)

Coefficient[t, x, 2] (* A266699 *)

LinearRecurrence[{2, 2, -1}, {0, -10, -12}, 30] (* Vincenzo Librandi, Jan 06 2016 *)

PROG

(MAGMA) I:=[0, -10, -12]; [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

(PARI) concat(0, Vec((-10*x+8*x^2)/(1-2*x-2*x^2+x^3) + O(x^100))) \\ Altug Alkan, Jan 07 2016

CROSSREFS

Cf. A265762, A266699.

Sequence in context: A324745 A267393 A248481 * A242508 A219917 A257039

Adjacent sequences:  A266697 A266698 A266699 * A266701 A266702 A266703

KEYWORD

sign,easy

AUTHOR

Clark Kimberling, Jan 05 2016

STATUS

approved

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Last modified May 30 10:28 EDT 2020. Contains 334724 sequences. (Running on oeis4.)