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

%I #28 Jan 25 2024 03:56:01

%S 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,

%T 19,20,20,20,21,22,23,24,24,24,25,26,27,28,28,28,29,30,31,32,32,32,33,

%U 34,35,36,36,36,37,38,39,40,40,40,41,42,43,44,44,44,45,46,47,48,48,48,49

%N Count, repeating 4n three times for n > 0.

%C A Chebyshev transform of A000975.

%C 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.

%C 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)).

%H Vincenzo Librandi, <a href="/A099479/b099479.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).

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

%F 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).

%F 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.

%F 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).

%F a(n) = floor((n+2)/3) + floor((n+3)/6) + floor((n+4)/6). - _Ridouane Oudra_, Jan 22 2024

%t 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 *)

%t LinearRecurrence[{2,-2,2,-2,2,-1},{0,1,2,3,4,4},80] (* _Harvey P. Dale_, Dec 11 2014 *)

%o (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

%o (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

%Y Cf. A099480.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 18 2004