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)).

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

Offset: 0

Author

Reinhard Zumkeller, Jul 14 2005

Keywords

• nonn
• easy
• frac

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.

Examples

			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, ...
		

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).

Crossrefs

Cf. A110091 (denominators).

Programs

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

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 *)

Sage

def A110090_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x+2*x^2-x^3)/(1-x^3)^2 ).list()
A110090_list(100) # G. C. Greubel, Jun 16 2021

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)