A172012 Expansion of (2-3*x)/(1-3*x-3*x^2) .

2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, 2326239, 8819442, 33437043, 126769455, 480619494, 1822166847, 6908359023, 26191577610, 99299809899, 376474162527, 1427321917278, 5411388239415, 20516130470079, 77782556128482, 294896059795683

Offset: 0

Author

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

Keywords

• nonn
• easy

Comments

The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.

The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.

Links

• Wikipedia, Lucas sequence: Specific names.
• Index entries for linear recurrences with constant coefficients, signature (3,3).

Programs

Mathematica

CoefficientList[Series[(2-3x)/(1-3x-3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,3},{2,3},31] (* Harvey P. Dale, Aug 24 2011 *)

Formula

a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).

L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).

a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). [Harvey P. Dale, Aug 24 2011]

a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

Extensions

Edited and extended by R. J. Mathar, Jan 23 2010