A052997 Expansion of (1+x-x^3)/((1-2*x)*(1-x^2)).

1, 3, 7, 14, 29, 58, 117, 234, 469, 938, 1877, 3754, 7509, 15018, 30037, 60074, 120149, 240298, 480597, 961194, 1922389, 3844778, 7689557, 15379114, 30758229, 61516458, 123032917, 246065834, 492131669, 984263338, 1968526677

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1075
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Programs

Maple

spec := [S,{S=Prod(Union(Sequence(Prod(Z,Z)),Z),Sequence(Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

Mathematica

f[s_List] := Block[{a = s[[-1]]}, Append[s, If[ OddQ@ Length@ s, 2a +1, 2a]]]; Join[{1},  Nest[f, {3}, 30]] (* or *)
CoefficientList[ Series[(1 + x - x^3)/(1 - 2x - x^2 + 2x^3), {x, 0, 30}], x] (* Robert G. Wilson v, Jul 20 2017 *)
LinearRecurrence[{2,1,-2},{1,3,7,14},40] (* Harvey P. Dale, May 27 2019 *)

Formula

G.f.: -(-x+x^3-1)/(-1+x^2)/(-1+2*x).

Recurrence: {a(0)=1, -2*a(n)-a(n+1)+a(n+2)-1, a(1)= 3, a(2)=7, a(3)=14}, 11/6*2^n + Sum(-1/6*(2 + _alpha)*_alpha^(-1-n), _alpha=RootOf(-1 + _Z^2))

a(n) = 2*a(n-1)+1 for even n, otherwise a(n) = 2*a(n-1), with a(0)=1, a(1)=3. [Bruno Berselli, Jun 19 2014]

3*a(n) = 11*2^(n-1)-A000034(n) for n>0. - R. J. Mathar, Feb 27 2019

Extensions

More terms from James Sellers, Jun 06 2000