OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-18).
FORMULA
a(n) = (3^n)*Stirling2(n+2, 2), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = 2*6^n - 3^n.
E.g.f.: (d^2/dx^2)((((exp(3*x)-1)/3)^2)/2!) = -exp(3*x) + 2*exp(6*x).
With leading zero, this is (6^n - 3^n)/3, the binomial transform of A016127 (with extra leading zero). - Paul Barry, Aug 20 2003
With leading zero, this is the fourth binomial transform of A001045, with a(n) = (2^n-1)(3^n/3 - 0^n/3) = A000225(n)*(A000244(n-1) - 0^n/3). - Paul Barry, Apr 28 2004
a(n) = Sum_{k=0..n} A100852(n,k). - Reinhard Zumkeller, Nov 20 2004
Sum_{k=1..n} 3^(k-1)*3^(n-k)*binomial(n, k). - Zerinvary Lajos, Sep 24 2006
a(n) = 9*a(n-1) - 18*a(n-2), n >= 2. - Vincenzo Librandi, Mar 14 2011
MATHEMATICA
Table[2*6^n -3^n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
CoefficientList[Series[1/((1-3x)(1-6x)), {x, 0, 40}], x] (* or *) LinearRecurrence[{9, -18}, {1, 9}, 40] (* Harvey P. Dale, Jul 07 2012 *)
PROG
(Sage) [lucas_number1(n, 9, 18) for n in range(1, 41)] # Zerinvary Lajos, Apr 23 2009
(PARI) Vec(1/(1-3*x)/(1-6*x)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
(Magma) [2*6^n -3^n: n in [0..40]]; // G. C. Greubel, Nov 14 2024
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More terms added by G. C. Greubel, Nov 14 2024
STATUS
approved