A099479 Count, repeating 4n three times for n > 0.

0, 1, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 17, 18, 19, 20, 20, 20, 21, 22, 23, 24, 24, 24, 25, 26, 27, 28, 28, 28, 29, 30, 31, 32, 32, 32, 33, 34, 35, 36, 36, 36, 37, 38, 39, 40, 40, 40, 41, 42, 43, 44, 44, 44, 45, 46, 47, 48, 48, 48, 49

Offset: 0

Author

Paul Barry, Oct 18 2004

Keywords

• easy
• nonn

Comments

A Chebyshev transform of A000975.

The denominator in the g.f. is 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + x^6, a version of the Jones polynomial of the knot 9_43.

The g.f. is the image of x/((1-x)*(1-x-2x^2)) under the Chebyshev transform A(x)->(1/(1+x^2))*A(x/(1+x^2)).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Crossrefs

Cf. A099480.

Programs

Magma

I:=[0,1,2,3,4,4]; [n le 6 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-2*Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..100]]; // Vincenzo Librandi, Sep 25 2013

Mathematica

CoefficientList[Series[x (1 + x^2)/((1 - x + x^2) (1 - x - x^3 + x^4)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 25 2013 *)
LinearRecurrence[{2,-2,2,-2,2,-1},{0,1,2,3,4,4},80] (* Harvey P. Dale, Dec 11 2014 *)

PARI

x='x+O('x^50); concat([0], vec(x*(1+x^2)/((1-x+x^2)*(1-x-x^3+x^4)))) \\ G. C. Greubel, Oct 10 2017

Formula

G.f.: x*(1+x^2)/((1-x+x^2)*(1-x-x^3+x^4)).

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6).

a(n) = -cos(Pi*2n/3 + Pi/3)/6 - sqrt(3)*sin(Pi*2n/3 + Pi/3)/18 - sqrt(3)*cos(Pi*n/3 + Pi/6)/6 - sin(Pi*n/3 + Pi/6)/2 + 2(n+1)/3.

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*(4*2^(n-1-2k)/3 - (-1)^n/6 - 1/2).

a(n) = floor((n+2)/3) + floor((n+3)/6) + floor((n+4)/6). - Ridouane Oudra, Jan 22 2024