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A097138 Convolution of 4^n and floor(n/2).

%I A097138 #37 May 09 2024 09:08:23
%S A097138 0,0,1,5,22,90,363,1455,5824,23300,93205,372825,1491306,5965230,
%T A097138 23860927,95443715,381774868,1527099480,6108397929,24433591725,
%U A097138 97734366910,390937467650,1563749870611,6254999482455,25019997929832
%N A097138 Convolution of 4^n and floor(n/2).
%C A097138 a(n+1) gives partial sums of A033114 and second partial sums of A015521.
%C A097138 Partial sums of 1/3*floor(4^n/5). - _Mircea Merca_, Dec 26 2010
%H A097138 Vincenzo Librandi, <a href="/A097138/b097138.txt">Table of n, a(n) for n = 0..200</a>
%H A097138 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-5,4).
%F A097138 G.f.: x^2/((1-x)*(1-4*x)*(1-x^2)).
%F A097138 a(n) = Sum_{k=0..n} floor((n-k)/2)4^k = Sum_{k=0..n} floor(k/2)*4^(n-k).
%F A097138 a(n) = 5*a(n-1) - 3*a(n-2) - 5*a(n-3) + 4*a(n-4).
%F A097138 From _Mircea Merca_, Dec 26 2010: (Start)
%F A097138 3*a(n) = round((16*4^n-30*n-25)/60) = floor((8*4^n-15*n-8)/30) = ceiling((8*4^n-15*n-17)/30) = round((8*4^n-15*n-8)/30).
%F A097138 a(n) = a(n-2)+(4^(n-1)-1)/3, n>1. (End)
%F A097138 a(n) = (4^(n+2)-30*n+9*(-1)^n-25)/180. - _Bruno Berselli_, Dec 27 2010
%F A097138 a(n) = (floor(4^(n+1)/15) - floor((n+1)/2))/3. - _Seiichi Manyama_, Dec 22 2023
%e A097138 a(3) = 1/3*floor(4^0/5)+1/3*floor(4^1/5)+1/3*floor(4^2/5) +1/3*floor(4^3/5) = 0 + 0 + 1 + 4 = 5.
%p A097138 A097138 := proc(n) (4^(n+2)-30*n+9*(-1)^n-25)/180 ; end proc: # _R. J. Mathar_, Jan 08 2011
%t A097138 LinearRecurrence[{5,-3,-5,4},{0,0,1,5},30] (* _Harvey P. Dale_, Sep 17 2017 *)
%o A097138 (Magma) [(4^(n+2)-30*n+9*(-1)^n-25)/180: n in [0..30]]; // _Vincenzo Librandi_, May 31 2011
%Y A097138 Column k=4 of A368296.
%Y A097138 Cf. A015521, A033114.
%K A097138 easy,nonn
%O A097138 0,4
%A A097138 _Paul Barry_, Jul 29 2004