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

0,1

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

Table of n, a(n) for n=0..25.

Vladimir V. Kruchinin and Maria Y. Perminova, Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.

Wikipedia, Lucas sequence: Specific names.

Index entries for linear recurrences with constant coefficients, signature (3,3).

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

E.g.f.: 2*exp(3*x/2)*cosh(sqrt(21)*x/2). - Stefano Spezia, Dec 21 2025

Mathematica

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

Crossrefs

Cf. A030195.

Sequence in context: A298409 A151369 A143885 * A323709 A338308 A047014

Adjacent sequences: A172009 A172010 A172011 * A172013 A172014 A172015

Keyword

nonn,easy,changed

Author

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

Extensions

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

Status

approved