login
A060925
a(n) = 2*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 4.
12
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
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
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