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%I #31 Sep 08 2022 08:45:52
%S 0,1,2,3,5,7,10,14,19,25,32,41,51,63,77,92,110,130,152,177,204,234,
%T 267,303,342,384,430,479,532,589,649,714,783,856,934,1016,1103,1195,
%U 1292,1394,1501,1614,1732,1856,1986,2121,2263,2411,2565,2726,2893
%N Partial sums of ceiling(n^2/15).
%C There are several sequences of integers of the form ceiling(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,8,10,11,12, 14,15,16,19,20,23,24).
%H Vincenzo Librandi, <a href="/A175846/b175846.txt">Table of n, a(n) for n = 0..10000</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_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,2,-2,1,-1,2,-1).
%F a(n) = round((2*n+1)*(n^2 + n + 28)/90).
%F a(n) = floor((n+1)*(2*n^2 + n + 56)/90).
%F a(n) = ceiling((2*n^3 + 3*n^2 + 57*n)/90).
%F a(n) = a(n-15) + (n+1)*(n-15) + 92.
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + a(n-7) - a(n-8) + 2*a(n-9) - a(n-10). - _R. J. Mathar_, Mar 11 2012
%F G.f.: x*(x+1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)/((x-1)^4*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)). - _Colin Barker_, Oct 26 2012
%e a(15) = 0 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 5 + 6 + 7 + 9 + 10 + 12 + 14 + 15 = 92.
%p seq(round((2*n+1)*(n^2+n+28)/90),n=0..50)
%o (Magma) [Round((2*n+1)*(n^2+n+28)/90): n in [0..60]]; // _Vincenzo Librandi_, Jun 22 2011
%o (PARI) a(n)=(n+1)*(2*n^2+n+56)\90 \\ _Charles R Greathouse IV_, Jul 06 2017
%K nonn,easy
%O 0,3
%A _Mircea Merca_, Dec 05 2010