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%I #94 Jun 22 2022 20:26:22
%S 0,1,2,0,-8,-16,0,64,128,0,-512,-1024,0,4096,8192,0,-32768,-65536,0,
%T 262144,524288,0,-2097152,-4194304,0,16777216,33554432,0,-134217728,
%U -268435456,0,1073741824,2147483648,0,-8589934592,-17179869184,0,68719476736,137438953472
%N Generalized Gaussian Fibonacci integers.
%C The sequence 0,1,-2,0,8,-16,... has g.f. x/(1+2*x-4*x^2), a(n) = 2^n*sin(2n*Pi/3)/sqrt(3) and is the inverse binomial transform of sin(sqrt(3)*x)/sqrt(3): 0,1,-3,0,9,...
%C a(n+1) is the Hankel transform of A100192. - _Paul Barry_, Jan 11 2007
%C a(n+1) is the trinomial transform of A010892: a(n+1) = Sum_{k=0..2n} trinomial(n,k)*A010892(k+1) where trinomial(n, k) = trinomial coefficients (A027907). - _Paul Barry_, Sep 10 2007
%C a(n+1) is the Hankel transform of A100067. - _Paul Barry_, Jun 16 2009
%C From _Paul Curtz_, Oct 04 2009: (Start)
%C 1) a(n) = A131577(n)*A128834(n).
%C 2) Binomial transform of 0,1,0,-3,0,9,0,-27, see A000244.
%C 3) Sequence is identical to every 2n-th difference divided by (-3)^n.
%C 4) a(3n) + a(3n+1) + a(3n+2) = (-1)^n*3*A001018(n) for n >= 1.
%C 5) For missing terms in a(n) see A013731 = 4*A001018. (End)
%C The coefficient of i of Q^n, where Q is the quaternion 1+i+j+k. Due to symmetry, also the coefficients of j and of k. - _Stanislav Sykora_, Jun 11 2012 [The coefficients of 1 are in A138230. - _Wolfdieter Lang_, Jan 28 2016]
%C With different signs, 0, 1, -2, 0, 8, -16, 0, 64, -128, 0, 512, -1024, ... is the Lucas U(-2,4) sequence. - _R. J. Mathar_, Jan 08 2013
%H Robert Israel, <a href="/A088138/b088138.txt">Table of n, a(n) for n = 0..3300</a>
%H Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, <a href="https://doi.org/10.1007/s00006-019-0969-9">On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis</a>, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-4).
%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F G.f.: x/(1-2*x+4*x^2).
%F E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3).
%F a(n) = 2*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1.
%F a(n) = ((1+i*sqrt(3))^n - (1-i*sqrt(3))^n)/(2*i*sqrt(3)).
%F a(n) = Im( (1+i*sqrt(3))^n/sqrt(3) ).
%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k+1)*(-3)^k.
%F From _Paul Curtz_, Oct 04 2009: (Start)
%F a(n) = a(n-1) + a(n-2) + 2*a(n-3).
%F a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). (End)
%F E.g.f.: exp(x)*sin(sqrt(3)*x)/sqrt(3) = G(0)*x^2 where G(k)= 1 + (3*k+2)/(2*x - 32*x^5/(16*x^4 - 3*(k+1)*(3*k+2)*(3*k+4)*(3*k+5)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 26 2012
%F G.f.: x/(1-2*x+4*x^2) = 2*x^2*G(0) where G(k)= 1 + 1/(2*x - 32*x^5/(16*x^4 - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 27 2012
%F a(n) = -2^(n-1)*Product_{k=1..n}(1 + 2*cos(k*Pi/n)) for n >= 1. - _Peter Luschny_, Nov 28 2019
%F a(n) = 2^(n-1) * U(n-1, 1/2), where U(n, x) is the Chebyshev polynomial of the second kind. - _Federico Provvedi_, Apr 24 2022
%p M:= <<1+I,1+I>|<I-1,1-I>>:
%p T:= <<-I/2,0>|<0,I/2>>:
%p seq(LinearAlgebra:-Trace(T.M^n),n=0..100); # _Robert Israel_, Jan 28 2016
%t Join[{a=0,b=1},Table[c=2*b-4*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)
%t LinearRecurrence[{2, -4}, {0, 1}, 40] (* _Vincenzo Librandi_, Jan 29 2016 *)
%t Table[2^(n-2)*((-1)^Quotient[n-1,3]+(-1)^Quotient[n,3]), {n,0,40}] (*_Federico Provvedi_,Apr 24 2022*)
%o (Sage) [lucas_number1(n,2,4) for n in range(0, 39)] # _Zerinvary Lajos_, Apr 23 2009
%o (PARI) /* lists powers of any quaternion) */
%o QuaternionToN(a,b,c,d,nmax) = {local (C);C = matrix(nmax+1,4);C[1,1]=1;for(n=2,nmax+1,C[n,1]=a*C[n-1,1]-b*C[n-1,2]-c*C[n-1,3]-d*C[n-1,4];C[n,2]=b*C[n-1,1]+a*C[n-1,2]+d*C[n-1,3]-c*C[n-1,4];C[n,3]=c*C[n-1,1]-d*C[n-1,2]+a*C[n-1,3]+b*C[n-1,4];C[n,4]=d*C[n-1,1]+c*C[n-1,2]-b*C[n-1,3]+a*C[n-1,4];);return (C);} /* _Stanislav Sykora_, Jun 11 2012 */
%o (PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-2*x+4*x^2))) \\ _G. C. Greubel_, Oct 22 2018
%o (PARI) a(n) = 2^(n-1)*polchebyshev(n-1, 2, 1/2); \\ _Michel Marcus_, May 02 2022
%o (Magma) I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 4*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%o (GAP) a:=[0,1];; for n in [3..40] do a[n]:=2*a[n-1]-4*a[n-2]; od; a; # _Muniru A Asiru_, Oct 23 2018
%Y Cf. A084102, A088137, A045873, A088139, A138230.
%K easy,sign
%O 0,3
%A _Paul Barry_, Sep 20 2003
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