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A052899 Expansion of g.f.: (1-2*x) / ((x-1)*(4*x^2+2*x-1)).

%I A052899 #89 Jul 02 2025 16:01:58
%S A052899 1,1,5,13,45,141,461,1485,4813,15565,50381,163021,527565,1707213,
%T A052899 5524685,17878221,57855181,187223245,605867213,1960627405,6344723661,
%U A052899 20531956941,66442808525,215013444813,695798123725,2251650026701,7286492548301,23579585203405,76305140600013
%N A052899 Expansion of g.f.: (1-2*x) / ((x-1)*(4*x^2+2*x-1)).
%C A052899 From _L. Edson Jeffery_, Apr 19 2011: (Start)
%C A052899 Let A be the unit-primitive matrix (see [Jeffery])
%C A052899 A = A_(10,4) =
%C A052899   (0 0 0 0 1)
%C A052899   (0 0 0 2 0)
%C A052899   (0 0 2 0 1)
%C A052899   (0 2 0 2 0)
%C A052899   (2 0 2 0 1).
%C A052899 Then a(n) = (1/5)*trace(A^n). (End)
%C A052899 a(n-1)+1 is the number of paths to reach a position outside a 4 X 4 chessboard after n steps, starting in one of the corners, when performing a walk with unit steps on the square lattice. - _Ruediger Jehn_, Oct 10 2024
%H A052899 Harvey P. Dale, <a href="/A052899/b052899.txt">Table of n, a(n) for n = 0..1000</a>
%H A052899 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=875">Encyclopedia of Combinatorial Structures 875</a>
%H A052899 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A052899 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-4).
%F A052899 Recurrence: {a(1)=1, a(0)=1, -4*a(n) - 2*a(n+1) + a(n+2) + 1 = 0}.
%F A052899 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 A052899 a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - _Gary W. Adamson_, Mar 01 2008
%F A052899 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 A052899 a(n) = (2*A087131(n)+1)/5. - _Bruno Berselli_, Apr 20 2011
%F A052899 a(n) = (2/5)*((1+sqrt(5))^n + (1-sqrt(5))^n + 1/2). - _Ruediger Jehn_, Sep 29 2024
%F A052899 E.g.f.: exp(x)*(1 + 4*cosh(sqrt(5)*x))/5. - _Stefano Spezia_, Oct 02 2024
%p A052899 spec := [S,{S=Sequence(Prod(Union(Sequence(Union(Z,Z)),Z,Z),Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052899 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 A052899 (Sage) from sage.combinat.sloane_functions import recur_gen2b
%o A052899 it = recur_gen2b(1,1,2,4, lambda n:-1)
%o A052899 [next(it) for i in range(1,28)] # _Zerinvary Lajos_, Jul 09 2008
%o A052899 (Magma) [(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // _Vincenzo Librandi_, Apr 20 2011
%o A052899 (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 A052899 Cf. A084057.
%K A052899 easy,nonn
%O A052899 0,3
%A A052899 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052899 More terms from _James Sellers_, Jun 08 2000