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A183895 Real part of a (-4,-4) Gaussian integer Somos-4 sequence. 4

%I

%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.

%F (End)

%t Table[Round[(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])], {n, 0, 30}] (* _G. C. Greubel_, Feb 21 2018 *)

%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(round((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)), ", ")) \\ _G. C. Greubel_, Feb 21 2018

%o (PARI) {a(n) = (-1)^(n + n\4) * (-2)^(n*(n+1)\4)}; /* _Michael Somos_, Mar 14 2020 */

%o (MAGMA) R:= RealField(); [Round((Sqrt(1/4-Sqrt(2)/8)*Sin(7*Pi(R)*n/4+3*Pi(R)/8) + Sqrt( Sqrt(2)/8+1/4)*Sin(5*Pi(R)*n/4+Pi(R)/8) +Sqrt(Sqrt(2)/8+1/4)*Cos(3* Pi(R)*n/4+3*Pi(R)/8) + Sqrt(1/4-Sqrt(2)/8)*Cos(Pi(R)*n/4+Pi(R)/8))*(-2)^Floor( Binomial( n+1,2)/2)): n in [0..30]]; // _G. C. Greubel_, Feb 21 2018

%Y Cf. A160637.

%K sign

%O 0,3

%A _Paul Barry_, Jan 07 2011

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Last modified December 5 12:04 EST 2020. Contains 338947 sequences. (Running on oeis4.)