OFFSET
0,2
COMMENTS
Binomial transform of A084059.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-3).
FORMULA
a(n) = ((3+sqrt(6))^n + (3-sqrt(6))^n)/2.
G.f.: (1-3*x)/(1-6*x+3*x^2).
E.g.f.: exp(3*x)*cosh(sqrt(6)*x).
a(n) = 3^n * Sum_{k=0..floor(n/2)} C(n, 2*k)*(2/3)^k. - Paul Barry, Sep 10 2005
Lim_{n -> oo} a(n)/a(n-1) = (3 + sqrt(6)) = 5.445489742... - Gary W. Adamson, Mar 19 2008
a(n) = Sum_{k=0..n} A147720(n,k)*3^k. - Philippe Deléham, Nov 15 2008
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(2*k-3)/(x*(2*k-1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
EXAMPLE
G.f. = 1 + 3*x + 15*x^2 + 81*x^3 + 441*x^4 + 2403*x^5 + 13095*x^6 + ...
MATHEMATICA
LinearRecurrence[{6, -3}, {1, 3}, 30] (* Harvey P. Dale, Feb 25 2014 *)
PROG
(PARI) {a(n) = if( n<0, 0, polsym(x^2 - 6*x + 3, n)[1+n] / 2)};
(Sage) [lucas_number2(n, 6, 3)/2 for n in range(0, 27)] # Zerinvary Lajos, Jul 08 2008
(Magma) [n le 2 select 3^(n-1) else 6*Self(n-1) -3*Self(n-2): n in [1..41]]; // G. C. Greubel, Oct 13 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 13 2003
STATUS
approved