A060925 a(n) = 2*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 4.

1, 4, 11, 34, 101, 304, 911, 2734, 8201, 24604, 73811, 221434, 664301, 1992904, 5978711, 17936134, 53808401, 161425204, 484275611, 1452826834, 4358480501, 13075441504, 39226324511, 117678973534, 353036920601

Offset

0,2

Comments

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=charpoly(A,2). - Milan Janjic, Jan 26 2010

Links

Harry J. Smith, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (2,3).

Formula

Row sums of Lucas convolution triangle A060922.

Inverse binomial transform of A003947. - Philippe Deléham, Jul 23 2005

a(n) = Sum_{m=0..n} A060922(n, m) = Sum_{j=1..n} (a(j-1)*A000204(n-j+1)) + A000204(n+1).

a(n) = (5*3^n - (-1)^n)/4.

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

a(2n) = 3*a(2n-1) - 1; a(2n+1) = 3*a(2n) + 1. - Philippe Deléham, Jul 23 2005

Binomial transform is A003947. - Paul Barry, May 19 2003

E.g.f.: (-exp(-x) + 5*exp(3*x))/4. - G. C. Greubel, Apr 06 2021

Maple

A060925:= n-> (5*3^n - (-1)^n)/4; seq(A060925(n), n=0..30); # G. C. Greubel, Apr 06 2021

Mathematica

f[n_]:=3/(n+2); x=2; Table[x=f[x]; Denominator[x], {n, 0, 5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 11 2010 *)
LinearRecurrence[{2, 3}, {1, 4}, 30] (* Harvey P. Dale, Mar 07 2014 *)

Prog

(PARI) {a(n) = (5*3^n - (-1)^n)/4};
vector(30, n, a(n-1)) \\ Harry J. Smith, Jul 19 2009 \\ modified by G. C. Greubel, Apr 06 2021
(Magma) [(5*3^n - (-1)^n)/4: n in [0..30]]; // G. C. Greubel, Apr 06 2021
(Sage) [(5*3^n - (-1)^n)/4 for n in (0..30)] # G. C. Greubel, Apr 06 2021

Crossrefs

Cf. A000204, A003947, A060922.

Sequence in context: A327548 A144791 A180305 * A027045 A243781 A227329

Adjacent sequences: A060922 A060923 A060924 * A060926 A060927 A060928

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 20 2001

Extensions

Recurrence, now used as definition, from Philippe Deléham, Jul 23 2005

Entry revised by N. J. A. Sloane, Sep 10 2006

Status

approved