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%I #21 Sep 08 2022 08:45:53
%S 0,0,0,0,1,2,3,5,7,10,14,18,23,29,36,44,53,63,75,88,102,118,135,154,
%T 175,197,221,247,275,305,337,371,408,447,488,532,578,627,679,733,790,
%U 850,913,979,1048,1120,1196,1275,1357,1443,1532
%N Partial sums of round(n^2/28).
%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="/A177277/b177277.txt">Table of n, a(n) for n = 0..905</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.
%F a(n) = round((2*n+1)*(2*n^2 + 2*n - 3)/336).
%F a(n) = floor((n+4)*(2*n^2 - 5*n + 18)/168).
%F a(n) = ceiling((n-3)*(2*n^2 + 9*n + 25)/168).
%F a(n) = round(n*(n+2)*(2*n-1)/168).
%F a(n) = a(n-28) + (n+1)*(n-28) + 275, n > 27.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-14) - 3*a(n-15) + 3*a(n-16) - a(n-17) with g.f. x^4*(1 - x + x^3 - x^4 + x^5 - x^7 + x^8) / ( (1+x)*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x-1)^4 ). - _R. J. Mathar_, Dec 12 2010
%e a(28) = 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 4 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 12 + 13 + 14 + 16 + 17 + 19 + 21 + 22 + 24 + 26 + 28 = 275.
%p seq(round(n*(n+2)*(2*n-1)/168),n=0..50)
%t Accumulate[Floor[Range[0,50]^2/28+1/2]] (* _Harvey P. Dale_, Feb 02 2012 *)
%o (Magma) [Floor((n+4)*(2*n^2-5*n+18)/168): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011
%Y Cf. A177100, A177116.
%K nonn,easy
%O 0,6
%A _Mircea Merca_, Dec 10 2010
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