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A117373 Expansion of (1 - 3x)/(1 - x + x^2). 12

%I

%S 1,-2,-3,-1,2,3,1,-2,-3,-1,2,3,1,-2,-3,-1,2,3,1,-2,-3,-1,2,3,1,-2,-3,

%T -1,2,3,1,-2,-3,-1,2,3,1,-2,-3,-1,2,3,1,-2,-3,-1,2,3,1,-2,-3,-1,2,3,1

%N Expansion of (1 - 3x)/(1 - x + x^2).

%C Row sums of number triangle A117372.

%C Periodic sequence with period {1, -2, -3, -1, 2, 3}. - _Philippe Deléham_, Nov 03 2008

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1).

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

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*(C(k,n-k) + 3*C(k, n-k-1)).

%F a(n) = (1/6)*(2*(n mod 6) - ((n+1) mod 6) - 3*((n+2) mod 6) - 2*((n+3) mod 6) + ((n+4) mod 6) + 3*((n+5) mod 6)), with n >= 1. - _Paolo P. Lava_, Apr 27 2007

%F a(n) = a(n-1) - a(n-2); a(0)=1, a(1)=-2. - _Philippe Deléham_, Nov 03 2008

%F a(n) = A010892(n) - 3*A010892(n-1). - _R. J. Mathar_, Sep 14 2013

%F a(n) = cos(n*Pi/3) - 5*sin(n*Pi/3)/sqrt(3). - _Andres Cicuttin_, Apr 06 2016

%F a(n) = ((n mod 3)^2 - 4*(n mod 3) + 1)*(-1)^floor(n/3). - _Luce ETIENNE_, Nov 18 2017

%t CoefficientList[Series[(1 - 3 x)/(1 - x + x^2), {x, 0, 200}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)

%o (PARI) Vec((1-3*x)/(1-x+x^2) + O(x^90)) \\ _Michel Marcus_, Apr 06 2016

%Y Cf. A010892, A117372.

%Y Cf. A010872 (n mod 3), A010875 (n mod 6).

%K easy,sign

%O 0,2

%A _Paul Barry_, Mar 10 2006

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Last modified November 18 14:59 EST 2019. Contains 329262 sequences. (Running on oeis4.)