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%I #25 Sep 08 2022 08:45:09
%S 0,1,6,19,60,181,546,1639,4920,14761,44286,132859,398580,1195741,
%T 3587226,10761679,32285040,96855121,290565366,871696099,2615088300,
%U 7845264901,23535794706,70607384119,211822152360,635466457081
%N Partial sums of A080925.
%C This is the sequence A(0,1;2,3;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. [_Wolfdieter Lang_, Oct 18 2010]
%H Vincenzo Librandi, <a href="/A080926/b080926.txt">Table of n, a(n) for n = 0..300</a>
%H Wolfdieter Lang, <a href="/A080926/a080926.pdf">Notes on certain inhomogeneous three term recurrences.</a> [_Wolfdieter Lang_, Oct 18 2010]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-3).
%F a(n) = Sum{i=0..n, Sum{k=1..i, Binomial(i, 2k-2)2^(2k-2)}}
%F G.f.: x*(1+3*x)/((1+x)*(1-x)*(1-3x)).
%F E.g.f.: (3*exp(3x)+exp(-x))/4-exp(x).
%F a(n) = (3*3^n+(-1)^n)/4-1.
%F a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) for n>2, a(0)=0, a(1)=1, a(2)=6. Observation by G. Detlefs. See my comment and link. [_Wolfdieter Lang_, Oct 18 2010]
%p a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+4 od: seq(a[n], n=0..33); # _Zerinvary Lajos_, Dec 14 2008
%t CoefficientList[Series[x (1 + 3 x) / ((1 + x) (1 - x) (1 - 3 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t LinearRecurrence[{3,1,-3},{0,1,6},30] (* _Harvey P. Dale_, Oct 02 2018 *)
%o (Magma) [(3*3^n+(-1)^n)/4-1: n in [0..30]]; // _Vincenzo Librandi_, Aug 06 2013
%K nonn,easy
%O 0,3
%A _Paul Barry_, Feb 26 2003
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