OFFSET
0,2
COMMENTS
Binomial transform of A001834. - Philippe Deléham, Nov 19 2009
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=-6.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=6.
Index entries for linear recurrences with constant coefficients, signature (6,-6).
FORMULA
a(n) = center term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 4 1 / 1 1 1]. M^n * [1 1 1] = [A083881(n) a(n) A083881(n)]. E.g., a(3) = 144 since M^3 * [1 1 1] = [54 144 54] = [A083881(3) a(3) A083881(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = (sqrt(6))^n*U(n, sqrt(6)/2).
G.f.: 1/(6*(x^2-x+1/6)).
Preceded by 0, this is the binomial transform of A001353. Its e.g.f. is then exp(3x)*sinh(sqrt(3)x)/sqrt(3). - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} A109466(n,k)*6^k. - Philippe Deléham, Oct 28 2008
a(n) = ((3+sqrt(3))^n - (3-sqrt(3))^n)/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
G.f.: A(x)= 1/(1-6*x+6*x^2) = G(0)/(1-3*x) where G(k) = 1 + 3*x/((1-3*x) - x*(1-3*x)/(x + (1-3*x)/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 28 2012
MATHEMATICA
Join[{a=1, b=6}, Table[c=6*b-6*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
PROG
(Sage) [lucas_number1(n, 6, 6) for n in range(1, 21)] # Zerinvary Lajos, Apr 22 2009
(PARI) a(n)=([0, 1; -6, 6]^n*[1; 6])[1, 1] \\ Charles R Greathouse IV, Jun 12 2015
(PARI) Vec(1/(6*x^2-6*x+1) + O(x^100)) \\ Colin Barker, Jun 15 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved