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%I #23 Mar 28 2024 09:04:06
%S 1,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,
%T -2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,
%U 2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2
%N a(n) = (-1)^n * 2 if n!=0, with a(0) = 1.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (-1).
%F Euler transform of length 2 sequence [-2, 1].
%F Moebius transform is length 2 sequence [-2, 4].
%F a(n) = -2*A033999(n) if n!=0.
%F G.f.: (1 - x) / (1 + x) = 1 / (1 + 2*x / (1 - x)) = 1 - 2*x / (1 + x).
%F E.g.f.: 2*exp(-x) - 1.
%F a(n) = a(-n) for all n in Z.
%F a(n) = A084100(2*n) = A084100(2*n + 1), if n>=0.
%F a(n) = (-1)^n * A040000(n).
%F a(2*n) = A040000(n).
%F Convolution inverse is A040000.
%e G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 2*x^7 + 2*x^8 - 2*x^9 + ...
%t a[ n_] := (-1)^n (2 - Boole[n == 0]);
%t PadRight[{1},120,{2,-2}] (* _Harvey P. Dale_, Jun 04 2019 *)
%o (PARI) {a(n) = (-1)^n * if(n, 2, 1)};
%o (Magma) [n eq 0 select 1 else 2*(-1)^n: n in [0..75]]; // _G. C. Greubel_, Jul 29 2018; Mar 28 2024
%o (SageMath) [2*(-1)^n -int(n==0) for n in range(76)] # _G. C. Greubel_, Mar 28 2024
%Y Cf. A033999, A040000, A084100.
%K sign,easy
%O 0,2
%A _Michael Somos_, Jan 05 2017
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