A080923 First differences of A003946.

1, 3, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616

Offset: 0

Author

Paul Barry, Feb 26 2003

Keywords

• easy
• nonn

Comments

Sum of consecutive pairs of elements of A025192.

The alternating sign sequence with g.f. (1-x^2)/(1+3x) gives the diagonal sums of A110168. - Paul Barry, Jul 14 2005

Let M = an infinite lower triangular matrix with the odd integers (1,3,5,...) in every column, with the leftmost column shifted up one row. Then A080923 = lim_{n->inf} M^n. - Gary W. Adamson, Feb 18 2010

a(n+1), n >= 0, with o.g.f. ((1-x^2)/(1-3*x)-1)/x = (3-x)/(1-3*x) provides the coefficients in the formal power series for tan(3*x)/tan(x) = (3-z)/(1-3*z) = Sum_{n>=0} a(n+1)*z^n, with z = tan(x)^2. Convergence holds for 0 <= z < 1/3, i.e., |x| < Pi/6, approximately 0.5235987758. For the numerator and denominator of this o.g.f. see A034867 and A034839, respectively. - Wolfdieter Lang, Jan 18 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (3).

Crossrefs

Essentially the same as A005051, A026097 and A083583.

Programs

Mathematica

CoefficientList[Series[(1 - x^2) / (1 - 3 x), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 05 2013 *)

Formula

G.f.: (1-x^2)/(1-3*x).

G.f.: 1/(1 - 3*x + x^2 - 3*x^3 + x^4 - 3*x^5 + ...). - Gary W. Adamson, Jan 06 2011

a(n) = 2^3*3^(n-2), n >= 2, a(0) = 1, a(1) = 3. - Wolfdieter Lang, Jan 18 2013