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%I #21 Sep 08 2022 08:46:06
%S 0,1,1,2,-3,-5,-8,13,21,34,-55,-89,-144,233,377,610,-987,-1597,-2584,
%T 4181,6765,10946,-17711,-28657,-46368,75025,121393,196418,-317811,
%U -514229,-832040,1346269,2178309,3524578,-5702887,-9227465,-14930352,24157817,39088169
%N a(n) = (-1)^floor( (n-1) / 3 ) * F(n), where F = Fibonacci.
%H Vincenzo Librandi, <a href="/A236191/b236191.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-4,0,0,1).
%F G.f.: (x + x^2 + 2*x^3 + x^4 - x^5) / (1 + 4*x^3 - x^6).
%F a(n) = -4 * a(n-3) + a(n-6). a(-n) = (-1)^floor( (n-2) / 3) * a(n) for all n in Z.
%F a(n) * a(n+3) = a(n+1)^2 - a(n+2)^2 for all n in Z.
%e G.f. = x + x^2 + 2*x^3 - 3*x^4 - 5*x^5 - 8*x^6 + 13*x^7 + 21*x^8 + ...
%t a[ n_] := (-1)^Quotient[n-1, 3] Fibonacci[n];
%t CoefficientList[Series[(x + x^2 + 2 x^3 + x^4 - x^5)/(1 + 4 x^3 - x^6), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jan 20 2014 *)
%o (PARI) {a(n) = (-1)^( (n-1) \ 3) * fibonacci( n)};
%o (Magma) I:=[0,1,1,2,-3,-5]; [n le 6 select I[n] else -4*Self(n-3)+Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Jan 20 2014
%Y Cf. A000045.
%K sign,easy
%O 0,4
%A _Michael Somos_, Jan 19 2014
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