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A181640 Partial sums of floor(n^2/5) (A118015). 1

%I #34 Sep 08 2022 08:45:54

%S 0,0,0,1,4,9,16,25,37,53,73,97,125,158,197,242,293,350,414,486,566,

%T 654,750,855,970,1095,1230,1375,1531,1699,1879,2071,2275,2492,2723,

%U 2968,3227,3500,3788,4092,4412,4748,5100,5469,5856,6261,6684,7125,7585,8065,8565

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

%H Vincenzo Librandi, <a href="/A181640/b181640.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 <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 a(n) = Sum_{k=0..n} floor(k^2/5).

%F a(n) = round((2*n^3 + 3*n^2 - 11*n - 6)/30).

%F a(n) = floor((2*n^3 + 3*n^2 - 11*n + 6)/30).

%F a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 18)/30).

%F a(n) = a(n-5) + (n-2)^2, n > 4.

%F From _Bruno Berselli_, Dec 15 2010: (Start)

%F G.f.: x^3*(1+x)/((1 + x + x^2 + x^3 + x^4)*(1-x)^4).

%F 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 a(5) = 9 = 0 + 0 + 0 + 1 + 3 + 5.

%p a(n):=round((2*n^(3)+3*n^(2)-11*n-6)/(30))

%o (Magma) [Floor((2*n^3+3*n^2-11*n+6)/30): n in [0..50]]; // _Vincenzo Librandi_, May 01 2011

%Y Cf. A118015.

%K nonn,easy

%O 0,5

%A _Mircea Merca_, Nov 18 2010

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)