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%I #29 Mar 20 2024 23:10:10
%S 1,-1,-2,8,32,-128,-1024,16384,262144,-4194304,-134217728,8589934592,
%T 549755813888,-35184372088832,-4503599627370496,1152921504606846976,
%U 295147905179352825856,-75557863725914323419136,-38685626227668133590597632,39614081257132168796771975168,40564819207303340847894502572032,-41538374868278621028243970633760768,-85070591730234615865843651857942052864,348449143727040986586495598010130648530944
%N Real part of a (-4,-4) Gaussian integer Somos-4 sequence.
%C Real part of the Hankel transform of A183893(n) + i*A183894(n).
%C A183895(n) + i*A183896(n) is a (-4,-4) Gaussian integer Somos-4 sequence.
%C This is a generalized Somos-4 sequence. - _Michael Somos_, Mar 14 2020
%H G. C. Greubel, <a href="/A183895/b183895.txt">Table of n, a(n) for n = 0..114</a>
%F a(n) = (sqrt(1/4 - sqrt(2)/8)*sin(7*Pi*n/4 + 3*Pi/8) + sqrt(sqrt(2)/8 + 1/4)*sin(5*Pi*n/4 + Pi/8) + sqrt(sqrt(2)/8 + 1/4)*cos(3*Pi*n/4 + 3*Pi/8) + sqrt(1/4 - sqrt(2)/8)*cos(Pi*n/4 + Pi/8))*(-2)^floor(binomial(n+1,2)/2).
%F From _Michael Somos_, Mar 14 2020: (Start)
%F a(n) = (-1)^(n + floor(n/4)) * A160637(n).
%F a(n) = a(-1-n) for all n in Z.
%F 0 = a(n)*a(n+4) + 6*a(n+1)*a(n+3) + 4*a(n+2)^2 for all n in Z.
%F 0 = a(n)*a(n+5) - 4*a(n+1)*a(n+4) for all n in Z. (End)
%F a(n) = (-1)^n * b(n+2), b() defined by 0 = b(n) * b(n+2) * b(n+3)^2 + b(n+4) * b(n+2) * b(n+1)^2 + b(n+1)^2 * b(n+3)^2, for n in N, all initial values +1. - _Helmut Ruhland_, Feb 22 2024
%t Table[(-1)^Floor[(n+1)/2]*2^Floor[n*(n+1)/4], {n,0,30}] (* _G. C. Greubel_, Feb 21 2018; Mar 18 2024 *)
%t a[ n_] := (-1)^(n + Quotient[n, 4])*(-2)^Quotient[n (n + 1), 4]; (* _Michael Somos_, Mar 14 2020 *)
%o (PARI) for(n=0,30, print1((-1)^((n+1)\2)*2^(n*(n+1)\4), ", ")) \\ _G. C. Greubel_, Feb 21 2018; Mar 18 2024
%o (PARI) {a(n) = (-1)^(n + n\4) * (-2)^(n*(n+1)\4)}; /* _Michael Somos_, Mar 14 2020 */
%o (Magma) [(-1)^Binomial(n+1,2)*2^Floor(n*(n+1)/4): n in [0..30]]; // _G. C. Greubel_, Feb 21 2018; Mar 18 2024
%o (SageMath) [(-1)^((n+1)//2)*2^(n*(n+1)//4) for n in range(31)] # _G. C. Greubel_, Mar 18 2024
%Y Cf. A160637, A183893, A183894, A183895, A183896.
%K sign
%O 0,3
%A _Paul Barry_, Jan 07 2011
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