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a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=0, a(1)=0, a(2)=1.
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%I #20 Sep 08 2022 08:46:17

%S 0,0,1,-1,-1,4,-3,-6,16,-7,-31,61,-6,-147,220,68,-655,739,639,-2772,

%T 2233,3950,-11188,5521,20805,-43035,6946,99929,-156856,-36056,449697,

%U -534441,-401009,1919588,-1652011,-2588174,7811784,-4287447,-13924295,30310973

%N a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=0, a(1)=0, a(2)=1.

%C Essentially the same as A077978. - _Georg Fischer_, Oct 02 2018

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-2,1).

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

%F Let P = (b-c)*(b-d), Q = (c-b)*(b-d), R = (d-b)*(d-c), (b, c, d) be the three roots of x^3 = 2*x^2 + x + 1, then a(n) = P^(-1)*b^(1-n) + Q^(-1)*c^(1-n) + R^(-1)*d^(1-n).

%F a(2*n) = -3*a(2*n-2) - 6*a(2*n-4) + a(2*n-6).

%t LinearRecurrence[{-1, -2, 1}, {0, 0, 1}, 50]

%t CoefficientList[Series[x^2/(1 + x + 2 x^2 - x^3), {x, 0, 39}], x] (* _Michael De Vlieger_, Aug 25 2016 *)

%o (Magma) I:=[0,0,1]; [n le 3 select I[n] else -Self(n-1)- 2*Self(n-2)+Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Aug 25 2016

%o (PARI) concat([0, 0], Vec(x^2/(1+x+2*x^2-x^3) + O(x^99))) \\ _Altug Alkan_, Aug 25 2016

%Y Cf. A077939, A276228.

%K sign,easy

%O 0,6

%A _G. C. Greubel_, Aug 24 2016