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A181640 Partial sums of floor(n^2/5) (A118015).

%I A181640 #43 Jun 14 2025 08:15:28
%S A181640 0,0,0,1,4,9,16,25,37,53,73,97,125,158,197,242,293,350,414,486,566,
%T A181640 654,750,855,970,1095,1230,1375,1531,1699,1879,2071,2275,2492,2723,
%U A181640 2968,3227,3500,3788,4092,4412,4748,5100,5469,5856,6261,6684,7125,7585,8065,8565
%N A181640 Partial sums of floor(n^2/5) (A118015).
%H A181640 Vincenzo Librandi, <a href="/A181640/b181640.txt">Table of n, a(n) for n = 0..1000</a>
%H A181640 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a> J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
%H A181640 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1,-3,3,-1).
%F A181640 a(n) = Sum_{k=0..n} floor(k^2/5).
%F A181640 a(n) = round((2*n^3 + 3*n^2 - 11*n - 6)/30).
%F A181640 a(n) = floor((2*n^3 + 3*n^2 - 11*n + 6)/30).
%F A181640 a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 18)/30).
%F A181640 a(n) = a(n-5) + (n-2)^2, n > 4.
%F A181640 From _Bruno Berselli_, Dec 15 2010: (Start)
%F A181640 G.f.: x^3*(1+x)/((1 + x + x^2 + x^3 + x^4)*(1-x)^4).
%F A181640 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8) for n > 7. (End)
%e A181640 a(5) = 9 = 0 + 0 + 0 + 1 + 3 + 5.
%p A181640 a(n):=round((2*n^(3)+3*n^(2)-11*n-6)/(30))
%o A181640 (Magma) [Floor((2*n^3+3*n^2-11*n+6)/30): n in [0..50]]; // _Vincenzo Librandi_, May 01 2011
%Y A181640 Cf. A118015.
%K A181640 nonn,easy
%O A181640 0,5
%A A181640 _Mircea Merca_, Nov 18 2010