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A080923 First differences of A003946. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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(a(n+1)*z^n,n=0..infty), 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).

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]

MAPLE

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL1, ZL2, ZL3), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=16):seq(count([S, {Q}, unlabelled], size=n), n=1..26); - Zerinvary Lajos, Mar 08 2008

with(finance):seq(ceil(futurevalue(8, 2, n)), n=-2..23); # [From Zerinvary Lajos, Mar 25 2009]

MATHEMATICA

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

CROSSREFS

Essentially the same as A005051, A026097 and A083583.

Sequence in context: A052855 A133787 * A118264 A006365 A178543 A188175

Adjacent sequences:  A080920 A080921 A080922 * A080924 A080925 A080926

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 26 2003

STATUS

approved

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Last modified May 25 16:15 EDT 2017. Contains 287039 sequences.