A355021 - OEIS

%I #34 Sep 06 2024 14:43:07

%S 1,-2,2,-5,6,-12,17,-30,46,-77,122,-200,321,-522,842,-1365,2206,-3572,

%T 5777,-9350,15126,-24477,39602,-64080,103681,-167762,271442,-439205,

%U 710646,-1149852,1860497,-3010350,4870846,-7881197,12752042,-20633240,33385281

%N a(n) = (-1)^n * A000032(n) - 1.

%C There are the partial sums of L(1) - L(2) + L(3) - L(4) + L(5) - ... .

%C Closely related (Fibonacci, A000045) partial sums of F(1) - F(2) + F(3) - F(4) + F(5) - ... are given by A355020.

%C Apart from signs, same as A098600 and A181716.

%H G. C. Greubel, <a href="/A355021/b355021.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 (0,2,-1).

%F a(n) = 2*a(n-2) - a(n-3) for n >= 3. [Corrected by _Georg Fischer_, Sep 30 2022]

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

%e a(0) = 1;

%e a(1) = 1 - 3 = -2;

%e a(2) = 1 - 3 + 4 = 2;

%e a(3) = 1 - 3 + 4 - 7 = -5.

%t f[n_] := Fibonacci[n]; g[n_] := LucasL[n];

%t f1 = Table[(-1)^n f[n] + 1, {n, 0, 40}]   (* A355020 *)

%t g1 = Table[(-1)^n g[n] - 1, {n, 0, 40}]   (* this sequence *)

%t LucasL[-Range[0, 50]] - 1 (* _G. C. Greubel_, Mar 17 2024 *)

%t LinearRecurrence[{0,2,-1},{1,-2,2},40] (* _Harvey P. Dale_, Sep 06 2024 *)

%o (Magma) [Lucas(-n) -1: n in [0..50]]; // _G. C. Greubel_, Mar 17 2024

%o (SageMath) [lucas_number2(-n,1,-1) -1 for n in range(51)] # _G. C. Greubel_, Mar 17 2024

%Y Cf. A000032, A000045, A098600, A181716, A355018, A355019, A355020.

%K sign,easy

%O 0,2

%A _Clark Kimberling_, Jun 21 2022