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A133585 Expansion of x - x^2*(2*x+1)*(x^2-2) / ( (x^2-x-1)*(x^2+x-1) ).

%I A133585 #24 Mar 04 2019 11:52:24
%S A133585 1,2,4,5,10,13,26,34,68,89,178,233,466,610,1220,1597,3194,4181,8362,
%T A133585 10946,21892,28657,57314,75025,150050,196418,392836,514229,1028458,
%U A133585 1346269,2692538,3524578,7049156,9227465,18454930,24157817
%N A133585 Expansion of x - x^2*(2*x+1)*(x^2-2) / ( (x^2-x-1)*(x^2+x-1) ).
%C A133585 A133585 is a companion to A133586.
%H A133585 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A133585 Equals the matrix-matrix-vector product A133566 * A133080 * A000045 (previous name).
%F A133585 For even-indexed terms, a(n) = F(n+1). For odd-indexed terms (n>1), a(n) = 2*a(n-1), A126358.
%e A133585 a(4) = F(5) = 5.
%e A133585 a(5) = 2*a(4) = 2*5 = 10.
%p A133585 A133585aux := proc(n,k)
%p A133585     add(A133566(n,j)*A133080(j,k),j=k..n) ;
%p A133585 end proc:
%p A133585 A000045 := proc(n)
%p A133585     combinat[fibonacci](n) ;
%p A133585 end proc:
%p A133585 A133585 := proc(n)
%p A133585     add(A133585aux(n,j)*A000045(j),j=0..n) ;
%p A133585 end proc: # _R. J. Mathar_, Jun 20 2015
%t A133585 CoefficientList[Series[1 - x (2 x + 1) (x^2 - 2)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 21 2015 *)
%t A133585 LinearRecurrence[{0,3,0,-1},{1,2,4,5,10},40] (* _Harvey P. Dale_, Mar 04 2019 *)
%o A133585 (PARI) a(n)=if(n>1,([0,1,0,0;0,0,1,0;0,0,0,1;-1,0,3,0]^(n-2)*[2;4;5;10])[1,1],1) \\ _Charles R Greathouse IV_, Jun 20 2015
%Y A133585 Cf. A133080, A000045, A133586.
%K A133585 nonn,easy
%O A133585 1,2
%A A133585 _Gary W. Adamson_, Sep 18 2007
%E A133585 Previous name corrected and new name from _R. J. Mathar_, Jun 20 2015