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A097137 Convolution of 3^n and floor(n/2).

%I A097137 #39 Dec 22 2023 10:37:11
%S A097137 0,0,1,4,14,44,135,408,1228,3688,11069,33212,99642,298932,896803,
%T A097137 2690416,8071256,24213776,72641337,217924020,653772070,1961316220,
%U A097137 5883948671,17651846024,52955538084,158866614264,476599842805
%N A097137 Convolution of 3^n and floor(n/2).
%C A097137 a(n+1) gives partial sums of A033113 and second partial sums of A015518(n+1). Binomial transform of {0,0,1,1,4,4,16,16,...}.
%C A097137 Partial sums of floor(3^n/8) = round(3^n/8). - _Mircea Merca_, Dec 28 2010
%H A097137 Vincenzo Librandi, <a href="/A097137/b097137.txt">Table of n, a(n) for n = 0..1000</a>
%H A097137 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,3).
%F A097137 G.f.: x^2/((1-x)^2*(1-3*x)*(1+x)).
%F A097137 a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) + 3*a(n-4).
%F A097137 a(n) = Sum_{k=0..n} floor((n-k)/2)*3^k = Sum_{k=0..n} floor(k/2)*3^(n-k).
%F A097137 From _Mircea Merca_, Dec 26 2010: (Start)
%F A097137 a(n) = round((3*3^n - 4*n - 4)/16) = floor((3*3^n - 4*n - 3)/16) = ceiling((3*3^n - 4*n - 5)/16) = round((3*3^n - 4*n - 3)/16).
%F A097137 a(n) = a(n-2) + (3^(n-1)-1)/2, n > 2. (End)
%F A097137 a(n) = (floor(3^(n+1)/8) - floor((n+1)/2))/2. - _Seiichi Manyama_, Dec 22 2023
%p A097137 A097137 := proc(n) add( floor(3^i/8),i=0..n) ; end proc:
%t A097137 CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)),{x,0,30}],x]  (* _Harvey P. Dale_, Mar 11 2011 *)
%o A097137 (Magma) [Round((3*3^n-4*n-4)/16): n in [0..30]]; // _Vincenzo Librandi_, Jun 25 2011
%o A097137 (PARI) my(x='x+O('x^30)); concat([0,0], Vec(x^2/((1-x)^2*(1-3*x)*(1+x)))) \\ _G. C. Greubel_, Jul 14 2019
%o A097137 (Sage) (x^2/((1-x)^2*(1-3*x)*(1+x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 14 2019
%o A097137 (GAP) a:=[0,0,1,4];; for n in [5..30] do a[n]:=4*a[n-1]-2*a[n-2]-4*a[n-3] +3*a[n-4]; od; a; # _G. C. Greubel_, Jul 14 2019
%Y A097137 Column k=3 of A368296.
%Y A097137 Cf. A033113.
%K A097137 nonn
%O A097137 0,4
%A A097137 _Paul Barry_, Jul 29 2004