A052986 Expansion of ( 1-2*x ) / ( (x-1)*(2*x^2+3*x-1) ).

1, 2, 7, 24, 85, 302, 1075, 3828, 13633, 48554, 172927, 615888, 2193517, 7812326, 27824011, 99096684, 352938073, 1257007586, 4476898903, 15944711880, 56787933445, 202253224094, 720335539171, 2565513065700, 9137210275441, 32542656957722, 115902391424047

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1060
• Index entries for linear recurrences with constant coefficients, signature (4,-1,-2).

Programs

Magma

I:=[1, 2, 7]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012

Maple

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

Mathematica

Join[{a=1,b=2},Table[c=3*b+2*a-1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
LinearRecurrence[{4,-1,-2},{1,2,7},40] (* Vincenzo Librandi, Jun 23 2012 *)

PARI

a(n) = round((1/4+(2^(-3-n)*((3-sqrt(17))^n*(-5+3*sqrt(17))+(3+sqrt(17))^n*(5+3*sqrt(17))))/sqrt(17))) \\ Colin Barker, Sep 02 2016

Formula

G.f.: (1-2*x)/(1-4*x+x^2+2*x^3).

Recurrence: {a(0)=1, a(1)=2, -2*a(n)-3*a(n+1)+a(n+2)+1=0}.

a(n) = Sum(-1/136*(-13-27*r+6*r^2)*r^(-1-n) where r=RootOf(1-4*_Z+_Z^2+2*_Z^3)).

a(n) = (1/4+(2^(-3-n)*((3-sqrt(17))^n*(-5+3*sqrt(17))+(3+sqrt(17))^n*(5+3*sqrt(17))))/sqrt(17)). - Colin Barker, Sep 02 2016

4*a(n) = 1+3*A007482(n)-2*A007482(n-1) - R. J. Mathar, Feb 27 2019

a(n)-a(n-1) = A007483(n-1). - R. J. Mathar, Jan 09 2025

Extensions

More terms from James Sellers, Jun 06 2000