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%I #46 Dec 28 2017 21:37:09
%S 0,0,0,1,3,6,10,15,22,31,42,55,71,90,112,137,165,197,233,273,317,366,
%T 420,479,543,612,687,768,855,948,1048,1155,1269,1390,1518,1654,1798,
%U 1950,2110,2279,2457,2644,2840,3045,3260,3485,3720,3965,4221,4488,4766
%N Partial sums of round(n^2/9).
%C Partial sums of A008740(n-3). [Corrected by _Gerald Hillier_, Dec 24 2017]
%C The round function, also called the nearest integer function, is defined here by round(x)=floor(x+1/2).
%C There are several sequences of integers of the form round(n^2/k) for whose partial sums we can establish identities as following (only for k = 2,...,9,11,12,13,16,17,19,20, 28,29,36,44).
%H Vincenzo Librandi, <a href="/A177100/b177100.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_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,1,-3,3,-1).
%F a(n) = round((n-1)*(n+2)*(2*n+1)/54);
%F a(n) = floor((n+3)*(2*n^2-3*n+6)/54);
%F a(n) = ceiling((n-2)*(2*n^2+7*n+11)/54);
%F a(n) = round((2*n^3+3*n^2-3*n)/54);
%F a(n) = a(n-9) + (n+1)*(n-9) + 31, n > 8.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-9) - 3*a(n-10) + 3*a(n-11) - a(n-12). - _R. J. Mathar_, Mar 11 2012
%F G.f.: x^3*(x+1)*(x^4-x^3+x^2-x+1)/((x-1)^4*(x^2+x+1)*(x^6+x^3+1)). - _Colin Barker_, Oct 10 2012
%e a(9) = 0+0+0+1+2+3+4+5+7+9 = 31.
%p seq(round((2*n^3+3*n^2-3*n)/54),n=0..50)
%t Accumulate[Round[Range[0,80]^2/9]] (* _Harvey P. Dale_, Apr 06 2017 *)
%o (PARI) a(n)=(n+3)*(2*n^2-3*n+6)\54 \\ _Charles R Greathouse IV_, Sep 28 2015
%Y Cf. A008740.
%K nonn,easy
%O 0,5
%A _Mircea Merca_, Dec 09 2010
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