%I #66 Dec 09 2022 07:10:34
%S -1,1,0,2,4,12,32,88,240,656,1792,4896,13376,36544,99840,272768,
%T 745216,2035968,5562368,15196672,41518080,113429504,309895168,
%U 846649344,2313089024,6319476736,17265131520,47169216512,128868696064,352075825152,961889042432,2627929735168
%N a(n+2) = 2*a(n+1) + 2*a(n); a(0) = -1, a(1) = 1.
%C a(n+1) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 1, 1]. - _R. J. Mathar_, Feb 04 2014
%C (A002605, a(.+1)) is the canonical basis of the space of linear recurrent sequences with signature (2, 2), i.e., any sequence s(n) = 2(s(n-1) + s(n-2)) is given by s = s(0)*A002605 + s(1)*a(.+1). - _M. F. Hasler_, Aug 06 2018
%H Reinhard Zumkeller, <a href="/A028860/b028860.txt">Table of n, a(n) for n = 0..1000</a>
%H Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=924">Encyclopedia of Combinatorial Structures 924</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 (2,2).
%F a(n) = 4*A028859(n-4), for n > 3.
%F From _R. J. Mathar_, Nov 27 2008: (Start)
%F G.f.: -(1 - 3*x)/(1 - 2*x - 2*x^2).
%F a(n) = 3*A002605(n-1) - A002605(n). (End)
%F a(n) = det A, where A is the Hessenberg matrix of order n+1 defined by: A[i,j] = p(j - i + 1) (i <= j), A[i,j] = -1 (i = j + 1), A[i,j] = 0 otherwise, with p(i) = fibonacci(2i - 4). - _Milan Janjic_, May 08 2010, edited by _M. F. Hasler_, Aug 06 2018
%F a(n) = (2*sqrt(3) - 3)/6*(1 + sqrt(3))^n - (2*sqrt(3) + 3)/6*(1 - sqrt(3))^n. - _Sergei N. Gladkovskii_, Jul 18 2012
%F a(n) = 2*A002605(n-2) for n >= 2. - _M. F. Hasler_, Aug 06 2018
%F E.g.f.: exp(x)*(2*sqrt(3)*sinh(sqrt(3)*x) - 3*cosh(sqrt(3)*x))/3. - _Franck Maminirina Ramaharo_, Nov 11 2018
%p seq(coeff(series((3*x-1)/(1-2*x-2*x^2), x,n+1),x,n),n=0..30); # _Muniru A Asiru_, Aug 07 2018
%t (With a different offset) M = {{0, 2}, {1, 2}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}] (* _Roger L. Bagula_, May 29 2005 *)
%t LinearRecurrence[{2,2},{-1,1},40] (* _Harvey P. Dale_, Dec 13 2012 *)
%t CoefficientList[Series[(-3 x + 1)/(2 x^2 + 2 x - 1), {x, 0, 27}], x] (* _Robert G. Wilson v_, Aug 07 2018 *)
%o (Haskell)
%o a028860 n = a028860_list !! n
%o a028860_list =
%o -1 : 1 : map (* 2) (zipWith (+) a028860_list (tail a028860_list))
%o -- _Reinhard Zumkeller_, Oct 15 2011
%o (PARI) apply( A028860(n)=([2,2;1,0]^n)[2,]*[1,-1]~, [0..30]) \\ 15% faster than (A^n*[1,-1]~)[2]. - _M. F. Hasler_, Aug 06 2018
%o (GAP) a:=[-1,1];; for n in [3..30] do a[n]:=2*a[n-1]+2*a[n-2]; od; a; # _Muniru A Asiru_, Aug 07 2018
%o (Magma) I:=[-1,1]; [n le 2 select I[n] else 2*Self(n-1)+2*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Aug 13 2018
%o (SageMath)
%o A028860 = BinaryRecurrenceSequence(2,2,-1,1)
%o [A028860(n) for n in range(51)] # _G. C. Greubel_, Dec 08 2022
%Y Cf. A002605, A026150, A030195, A080040, A083337, A106435, A108898, A125145.
%K sign,easy
%O 0,4
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Apr 11 2009
%E Edited and initial values added in definition by _M. F. Hasler_, Aug 06 2018