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A110090 Numerators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)). 2
0, 1, 2, 2, 1, 4, 4, 1, 6, 6, 1, 8, 8, 1, 10, 10, 1, 12, 12, 1, 14, 14, 1, 16, 16, 1, 18, 18, 1, 20, 20, 1, 22, 22, 1, 24, 24, 1, 26, 26, 1, 28, 28, 1, 30, 30, 1, 32, 32, 1, 34, 34, 1, 36, 36, 1, 38, 38, 1, 40, 40, 1, 42, 42, 1, 44, 44, 1, 46, 46, 1, 48, 48, 1, 50, 50, 1, 52, 52, 1, 54, 54 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

r(n) = a(n) / A110091(n);

a(n) = (A110091(n-1)+A110091(n-2)) / GCD(a(n-1)+a(n-2), A110091(n-1)+A110091(n-2));

r --> 1.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

FORMULA

a(3*k) = 2*k, a(3*k+1) = 1, a(3*k+2) = 2*(k+1).

a(n) = (1/9)*(4*n + 5 + (4*n-4)*cos((2*n+1)*Pi/3) + sqrt(3)*sin(2*(n-1)*Pi/3) - sqrt(3)*sin(4*(n-1)*Pi/3)). - Wesley Ivan Hurt, Sep 25 2017

From Vincenzo Librandi, Sep 27 2017: (Start)

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

a(n) = 2*a(n-3) - a(n-6) for n>8. (End)

EXAMPLE

First terms of r: 0 1 2 2/3 1 4/3 4/5 1 6/5 6/7 1 8/7 ...:

r(2)=(1+1)/(1+0)=2, r(3)=(1+1)/(2+1)=2/3, r(4)=(3+1)/(2+2)=1,

r(5)=(1+3)/(1+2)=4/3, r(6)=(3+1)/(4+1)=4/5, ...

MATHEMATICA

Join[{0, 1}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {2, 2, 1, 4, 4, 1}, 100]] (* or *) CoefficientList[Series[x  (1 + 2 x + 2 x^2 - x^3) / ((1 - x)^2 (1 + x + x^2)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2017 *)

PROG

(MAGMA) I:=[0, 1, 2, 2, 1, 4, 4, 1]; [n le 8 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..100]]; // Vincenzo Librandi, Sep 27 2017

CROSSREFS

Sequence in context: A334770 A274622 A138189 * A196831 A092848 A128111

Adjacent sequences:  A110087 A110088 A110089 * A110091 A110092 A110093

KEYWORD

nonn,easy,frac

AUTHOR

Reinhard Zumkeller, Jul 14 2005

STATUS

approved

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Last modified June 17 05:45 EDT 2021. Contains 345080 sequences. (Running on oeis4.)