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%I #60 Sep 08 2022 08:44:59
%S 1,1,5,13,45,141,461,1485,4813,15565,50381,163021,527565,1707213,
%T 5524685,17878221,57855181,187223245,605867213,1960627405,6344723661,
%U 20531956941,66442808525,215013444813,695798123725,2251650026701
%N Expansion of g.f.: ( 1-2*x ) / ((x-1)*(4*x^2+2*x-1)).
%C From _L. Edson Jeffery_, Apr 19 2011: (Start)
%C Let A be the unit-primitive matrix (see [Jeffery])
%C A = A_(10,4) =
%C (0 0 0 0 1)
%C (0 0 0 2 0)
%C (0 0 2 0 1)
%C (0 2 0 2 0)
%C (2 0 2 0 1).
%C Then a(n) = (1/5)*trace(A^n). (End)
%H Harvey P. Dale, <a href="/A052899/b052899.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=875">Encyclopedia of Combinatorial Structures 875</a>
%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-4).
%F Recurrence: {a(1)=1, a(0)=1, -4*a(n) - 2*a(n+1) + a(n+2) + 1 = 0}.
%F a(n) = Sum(-1/25*(-1-8*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-2*_Z^2+4*_Z^3)).
%F a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - _Gary W. Adamson_, Mar 01 2008
%F a(n) = (1/5) * Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1), x_k=2*cos((2*k-1)*Pi/10). Also, a(n)/a(n-1) -> spectral radius of matrix A_(10,4) above. - _L. Edson Jeffery_, Apr 19 2011
%F a(n) = (2*A087131(n)+1)/5. - _Bruno Berselli_, Apr 20 2011
%p spec := [S,{S=Sequence(Prod(Union(Sequence(Union(Z,Z)),Z,Z),Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t CoefficientList[Series[(1-2x)/((x-1)(4x^2+2x-1)),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,-4},{1,1,5},40] (* _Harvey P. Dale_, Jul 10 2017 *)
%o (Sage) from sage.combinat.sloane_functions import recur_gen2b
%o it = recur_gen2b(1,1,2,4, lambda n:-1)
%o [next(it) for i in range(1,28)] # _Zerinvary Lajos_, Jul 09 2008
%o (Magma) [(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // _Vincenzo Librandi_, Apr 20 2011
%o (Maxima) makelist(coeff(taylor((1-2*x)/(1-3*x-2*x^2+4*x^3),x,0,n),x,n),n,0,25); /* _Bruno Berselli_, May 30 2011 */
%Y Cf. A084057.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E More terms from _James A. Sellers_, Jun 08 2000
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