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%I #40 Jan 01 2024 18:41:23
%S 1,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,
%T -3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,
%U -3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3,0,3,3,0,-3,-3
%N Expansion of (1+x)^3/(1+x^3).
%C Partial sums are A084104.
%H G. C. Greubel, <a href="/A084103/b084103.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1).
%F G.f.: (1+x)^3/(1+x^3).
%F a(n) = Sum_{k=0..n} binomial(2n-k-1, k)(-1)^k*3(n-k). - _Paul Barry_, Jan 21 2005
%F a(0) = 1 and a(n) = 2*sqrt(3)*sin(n*Pi/3). - _N-E. Fahssi_, Mar 04 2010
%F Euler transform of length 6 sequence [3, -3, -1, 0, 0, 1]. - _Michael Somos_, Feb 13 2011
%F a(n) = -a(-n) = 3 * A128834(n) except a(0) = 1. - _Michael Somos_, Feb 13 2011
%F a(n) = 3*(n^2 mod 3)*(-1)^floor(n/3), n>0. - _Wesley Ivan Hurt_, May 15 2015
%F The periodic sequence b(n) = a(n+1) has the o.g.f. 3 + G(x) = 3 + 3x(1-x) / (1-x(1-x)) = 3 + 3 L(Cinv(x)) = 3 + 3 x - 3 x^3 - 3 x^4 + ... , where L(x) = x/(1-x) with inverse Linv(x) = x/(1+x) and Cinv(x) = x(1-x), the inverse of the o.g.f. for the shifted Catalan numbers of A000108, C(x) = (1-sqrt(1-4x))/2. Then Ginv(x) = C(Linv(x/3)) = [1 - sqrt[1-4x/(3+x)]]/2. Cf. A267633. - _Tom Copeland_, Jan 25 2016
%e 1 + 3*x + 3*x^2 - 3*x^4 - 3*x^5 + 3*x^7 + 3*x^8 - 3*x^10 - 3*x^11 + ...
%p 1, seq(op((-1)^i*[3, 3, 0]), i=0..20); # _Robert Israel_, May 17 2015
%t CoefficientList[Series[(1 + x)^3/(1 + x^3), {x, 0, 100}], x] (* _Vincenzo Librandi_, May 16 2015 *)
%t Join[{1}, LinearRecurrence[{1,-1}, {3,3}, 30]] (* _G. C. Greubel_, Jan 15 2018 *)
%o (PARI) {a(n) = (n==0) + [0, 3, 3, 0, -3, -3][n%6 + 1]} /* _Michael Somos_, Feb 13 2011 */
%o (PARI) {a(n) = (n==0) - 3 * (-1)^n * kronecker(-3, n)} /* _Michael Somos_, Feb 13 2011 */
%o (Magma) I:=[1,3,3]; [n le 3 select I[n] else Self(n-1)-Self(n-2): n in [1..100]]; // _Vincenzo Librandi_, May 16 2015
%Y Cf. A000108, A084104, A267633.
%K easy,sign
%O 0,2
%A _Paul Barry_, May 15 2003