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A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

%I A226328 #46 Feb 12 2024 10:17:40
%S A226328 1,-2,2,1,2,6,9,14,26,41,66,110,177,286,466,753,1218,1974,3193,5166,
%T A226328 8362,13529,21890,35422,57313,92734,150050,242785,392834,635622,
%U A226328 1028457,1664078,2692538,4356617,7049154,11405774,18454929,29860702,48315634,78176337
%N A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).
%C A226328 a(n+1)/a(n) -> the golden ratio, A001622.
%C A226328 a(3*n)+a(3*n+1)+a(3*n+2) = 1,9,49,217,929,... = b(n), and b(n+1)-b(n) = 8*A015448(n+1).
%H A226328 Vincenzo Librandi, <a href="/A226328/b226328.txt">Table of n, a(n) for n = 0..1000</a>
%H A226328 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).
%F A226328 a(n) = F(n-3) + F(n) - A010872(n+1).
%F A226328 a(n+3) = a(n) + 4*F(n).
%F A226328 G.f.: (2*x^4+3*x^2-3*x+1)/( (x-1)*(x^2+x-1)*(1+x+x^2) ). [_Charles R Greathouse IV_, Jun 04 2013]
%F A226328 a(n) = A057078(n+1) +2*A212804(n) -1. - _R. J. Mathar_, Jun 26 2013
%e A226328 a(2)=-2+1+3=2, a(3)=2-2+1=1, a(4)=1+2-1=2, a(5)=2+1+3=6.
%e A226328 a(0)=F(-3)+F(n)-1=2+0-1=1,  a(1)=-1+1-2=-2, a(2)=1+1-0=2.
%e A226328 a(3)=1+4*0=1, a(4)=-2+4*1=2, a(5)=2+4*1=6, a(6)=1+4*2=9.
%t A226328 CoefficientList[Series[(2 x^4 + 3 x^2 - 3 x + 1) / (x^5 + x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 05 2013 *)
%t A226328 LinearRecurrence[{1, 1, 1, -1, -1}, {1, -2, 2, 1, 2}, 40] (* _Hugo Pfoertner_, Feb 12 2024 *)
%o A226328 (PARI) Vec((2*x^4+3*x^2-3*x+1)/(x^5+x^4-x^3-x^2-x+1)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 04 2013
%o A226328 (Magma) I:=[1,-2,2,1,2]; [n le 5 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4)-Self(n-5): n in [1..40]]; // _Vincenzo Librandi_, Jun 05 2013
%Y A226328 Cf. A156279, A022120, A022353.
%K A226328 sign,easy,less
%O A226328 0,2
%A A226328 _Paul Curtz_, Jun 04 2013
%E A226328 a(23) corrected by _Charles R Greathouse IV_, Jun 04 2013
%E A226328 More terms from _Bruno Berselli_, Jun 04 2013