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%I #42 Jan 11 2024 00:49:05
%S 0,1,4,10,16,4,-80,-344,-896,-1520,-704,6304,29440,79936,143104,92800,
%T -487424,-2506496,-7101440,-13366784,-10858496,36766720,212217856,
%U 628271104,1239777280,1189482496,-2680733696,-17859829760,-55354916864,-114260688896
%N a(n) = 4*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.
%C For the quaternion Q = 2+j+k, Q^n = r(n) + a(n)*(j+k). The sequence of real-parts r(n) is A266046. - _Stanislav Sykora_, Dec 20 2015
%H Stanislav Sykora, <a href="/A190965/b190965.txt">Table of n, a(n) for n = 0..1000</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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6).
%F G.f.: x/(1-4*x+6*x^2). - _Philippe Deléham_, Oct 12 2011
%F 2*a(n)^2 + A266046(n)^2 = 6^n. - _Stanislav Sykora_, Dec 20 2015
%F From _G. C. Greubel_, Jan 10 2024: (Start)
%F a(n) = 6^((n-1)/2)*ChebyshevU(n-1, sqrt(2/3)).
%F E.g.f.: (1/sqrt(2))*exp(2*x)*sin(sqrt(2)*x). (End)
%t LinearRecurrence[{4,-6}, {0,1}, 50]
%o (PARI) a(n)=([0,1;0,0]*[0,-6;1,4]^n)[1,1] \\ _Charles R Greathouse IV_, May 31 2011
%o (Magma) [n le 2 select n-1 else 4*Self(n-1) -6*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Jan 10 2024
%o (SageMath)
%o A190965=BinaryRecurrenceSequence(4,-6,0,1)
%o [A190965(n) for n in range(41)] # _G. C. Greubel_, Jan 10 2024
%Y Cf. A190958 (index to generalized Fibonacci sequences).
%Y Cf. A088137 (Inv. Bin. Trans.), A168175, A213421, A266046.
%K sign,easy
%O 0,3
%A _Vladimir Joseph Stephan Orlovsky_, May 24 2011
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