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

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000204, A003947, A060922.

Programs

Magma

[(5*3^n - (-1)^n)/4: n in [0..30]]; // 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 *)

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

Sage

[(5*3^n - (-1)^n)/4 for n in (0..30)] # G. C. Greubel, Apr 06 2021

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

Extensions

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

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