A233828 - OEIS

%I #13 Sep 08 2022 08:46:06

%S -1,1,1,3,9,25,71,201,569,1611,4561,12913,36559,103505,293041,829651,

%T 2348889,6650121,18827671,53304473,150914409,427265435,1209664161,

%U 3424773601,9696140959,27451493281,77720042081,220039211683,622970000809,1763738467065

%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="/A233828/b233828.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 + 3*x + x^2) / (1 - 2*x - 2*x^2 - x^3).

%F a(n+2) = A101168(n). a(-n) = A233831(n).

%F a(n) - a(n-1) = -2 * (-1)^n * A078054(n-3).

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

%e G.f. = -1 + x + x^2 + 3*x^3 + 9*x^4 + 25*x^5 + 71*x^6 + 201*x^7 + 569*x^8 + ...

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

%o (PARI) {a(n) = if( n<0, polcoeff( (-1 -x + x^2) / (1 + 2*x + 2*x^2 - x^3) + x * O(x^-n), -n), polcoeff( (-1 + 3*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+3*x+x^2)/(1-2*x-2*x^2-x^3))); // _G. C. Greubel_, Aug 07 2018

%Y Cf. A078004, A078054, A101168, A233831.

%K sign

%O 0,4

%A _Michael Somos_, Dec 16 2013