A233831 - OEIS

%I #13 Nov 28 2024 16:44:00

%S -1,1,1,-5,9,-7,-9,41,-71,51,81,-335,559,-367,-719,2731,-4391,2601,

%T 6311,-22215,34409,-18077,-54879,180321,-268961,122401,473441,

%U -1460645,2096809,-798887,-4056489,11807561,-16301031,4930451,34548721,-95259375,126351759

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

%H G. C. Greubel, <a href="/A233831/b233831.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(-n) = A233828(n).

%F a(n) - a(n-1) = -2 * (-1)^n * A078004(n).

%F a(n)^2 - a(n-1) * a(n+1) = -2 * (-1)^n * A078054(n-1).

%e G.f. = -1 + x + x^2 - 5*x^3 + 9*x^4 - 7*x^5 - 9*x^6 + 41*x^7 - 71*x^8 + ...

%t CoefficientList[Series[(-1-x+x^2)/(1+2*x+2*x^2-x^3), {x, 0, 50}], x] (* _G. C. Greubel_, Aug 07 2018 *)

%t LinearRecurrence[{-2,-2,1},{-1,1,1},40] (* _Harvey P. Dale_, Nov 28 2024 *)

%o (PARI) {a(n) = if( n<0, polcoeff( (-1 +3*x + x^2) / (1 - 2*x - 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( (-1 - x + x^2) / (1 + 2*x + 2*x^2 - x^3) + x * O(x^n), n))}

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((-1-x+x^2)/(1+2*x+2*x^2-x^3))); // _G. C. Greubel_, Aug 07 2018

%Y Cf. A078004, A078054, A233828.

%K sign

%O 0,4

%A _Michael Somos_, Dec 16 2013