This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A190965 #54 Jan 04 2025 12:39:13
%S A190965 0,1,4,10,16,4,-80,-344,-896,-1520,-704,6304,29440,79936,143104,92800,
%T A190965 -487424,-2506496,-7101440,-13366784,-10858496,36766720,212217856,
%U A190965 628271104,1239777280,1189482496,-2680733696,-17859829760,-55354916864,-114260688896
%N A190965 a(n) = 4*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.
%C A190965 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 A190965 Stanislav Sykora, <a href="/A190965/b190965.txt">Table of n, a(n) for n = 0..1000</a>
%H A190965 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 A190965 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6).
%F A190965 G.f.: x/(1-4*x+6*x^2). - _Philippe Deléham_, Oct 12 2011
%F A190965 2*a(n)^2 + A266046(n)^2 = 6^n. - _Stanislav Sykora_, Dec 20 2015
%F A190965 From _G. C. Greubel_, Jan 10 2024: (Start)
%F A190965 a(n) = 6^((n-1)/2)*ChebyshevU(n-1, sqrt(2/3)).
%F A190965 E.g.f.: (1/sqrt(2))*exp(2*x)*sin(sqrt(2)*x). (End)
%F A190965 a(n) = (i*sqrt(2)/4)*((2 - i*sqrt(2))^n - (2 + i*sqrt(2))^n), where i = sqrt(-1). - _Taras Goy_, Jan 03 2025
%p A190965 w := I*sqrt(2): a := n -> (w/4)*((2 - w)^n - (2 + w)^n):
%p A190965 seq(simplify(a(n)), n = 0..20); # (after _Taras Goy_), _Peter Luschny_, Jan 03 2025
%t A190965 LinearRecurrence[{4,-6}, {0,1}, 50]
%o A190965 (PARI) a(n)=([0,1;0,0]*[0,-6;1,4]^n)[1,1] \\ _Charles R Greathouse IV_, May 31 2011
%o A190965 (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 A190965 (SageMath)
%o A190965 A190965=BinaryRecurrenceSequence(4,-6,0,1)
%o A190965 [A190965(n) for n in range(41)] # _G. C. Greubel_, Jan 10 2024
%Y A190965 Cf. A190958 (index to generalized Fibonacci sequences).
%Y A190965 Cf. A088137 (Inv. Bin. Trans.), A168175, A213421, A266046.
%K A190965 sign,easy
%O A190965 0,3
%A A190965 _Vladimir Joseph Stephan Orlovsky_, May 24 2011