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%I #36 Aug 14 2017 02:44:22
%S 1,3,8,24,72,216,648,1944,5832,17496,52488,157464,472392,1417176,
%T 4251528,12754584,38263752,114791256,344373768,1033121304,3099363912,
%U 9298091736,27894275208,83682825624,251048476872,753145430616
%N First differences of A003946.
%C Sum of consecutive pairs of elements of A025192.
%C The alternating sign sequence with g.f. (1-x^2)/(1+3x) gives the diagonal sums of A110168. - _Paul Barry_, Jul 14 2005
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A080923/b080923.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (3).
%F G.f.: (1-x^2)/(1-3*x).
%F G.f.: 1/(1 - 3*x + x^2 - 3*x^3 + x^4 - 3*x^5 + ...). - _Gary W. Adamson_, Jan 06 2011
%F a(n) = 2^3*3^(n-2), n >= 2, a(0) = 1, a(1) = 3. - _Wolfdieter Lang_, Jan 18 2013
%t CoefficientList[Series[(1 - x^2) / (1 - 3 x), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 05 2013 *)
%Y Essentially the same as A005051, A026097 and A083583.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 26 2003
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