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%I #50 Apr 21 2023 16:27:23
%S 1,-5,21,-89,377,-1597,6765,-28657,121393,-514229,2178309,-9227465,
%T 39088169,-165580141,701408733,-2971215073,12586269025,-53316291173,
%U 225851433717,-956722026041,4052739537881,-17167680177565,72723460248141,-308061521170129,1304969544928657
%N A transform of the Fibonacci numbers.
%C The g.f. is the transform of the g.f. of A000045 under the mapping G(x) -> (-1/(1+x))*G((x-1)/(x+1)). In general this mapping transforms x/(1-k*x-k*x^2) into (1-x)/(1 + 2(k+1)*x - (2*k-1)*x^2).
%C Pisano period lengths: 1, 1, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20, ... - _R. J. Mathar_, Aug 10 2012
%H G. C. Greubel, <a href="/A099843/b099843.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-4,1).
%F G.f.: (1-x)/(1+4*x-x^2).
%F a(n) = (sqrt(5)-2)^n * (1/2 - 3*sqrt(5)/10) + (-sqrt(5)-2)^n * (1/2 + 3*sqrt(5)/10).
%F a(n) = (-1)^n*Fibonacci(3*n+2).
%F a(n) = -4*a(n-1) + a(n-2), a(0)=1, a(1)=-5. - _Philippe Deléham_, Nov 03 2008
%F a(n) = (-1)^n*(A001076(n) + A001076(n+1)). - _R. J. Mathar_, Aug 10 2012
%F a(n) = (-1)^n*A015448(n+1). - _R. J. Mathar_, May 07 2019
%p a:= n-> (<<0|1>, <1|-4>>^n.<<1, -5>>)[1,1]:
%p seq(a(n), n=0..24); # _Alois P. Heinz_, Apr 21 2023
%t CoefficientList[Series[(1-x)/(1+4*x-x^2), {x,0,30}], x] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%t LinearRecurrence[{-4,1},{1,-5},30] (* _Harvey P. Dale_, Aug 13 2015 *)
%o (Magma) [(-1)^n*Fibonacci(3*n+2): n in [0..40]]; // _G. C. Greubel_, Apr 20 2023
%o (SageMath) [(-1)^n*fibonacci(3*n+2) for n in range(41)] # _G. C. Greubel_, Apr 20 2023
%Y Cf. A000045, A001076, A015448, A099842.
%Y Cf. A084326 (shifted unsigned inverse binomial transform), A152174 (binomial transform).
%K easy,sign
%O 0,2
%A _Paul Barry_, Oct 27 2004
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