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A177176 Partial sums of round(n^2/13).

%I A177176 #25 Sep 08 2022 08:45:53
%S A177176 0,0,0,1,2,4,7,11,16,22,30,39,50,63,78,95,115,137,162,190,221,255,292,
%T A177176 333,377,425,477,533,593,658,727,801,880,964,1053,1147,1247,1352,1463,
%U A177176 1580,1703,1832,1968,2110,2259,2415,2578,2748,2925,3110,3302
%N A177176 Partial sums of round(n^2/13).
%C A177176 The round function is defined here by round(x) = floor(x + 1/2).
%H A177176 Vincenzo Librandi, <a href="/A177176/b177176.txt">Table of n, a(n) for n = 0..10000</a>
%H A177176 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 A177176 a(n) = round(n*(n+1)*(2*n+1)/78).
%F A177176 a(n) = floor((n+3)*(2*n^2 - 3*n + 10)/78).
%F A177176 a(n) = ceiling((n-2)*(2*n^2 + 7*n + 15)/78).
%F A177176 a(n) = a(n-13) + (n+1)*(n-13) + 63, n > 12.
%F A177176 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-13) - 3*a(n-14) + 3*a(n-15) - a(n-16) with g.f. x^3*(1+x)*(x^2 - x + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1) / ( (x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x-1)^4 ). - _R. J. Mathar_, Dec 13 2010
%e A177176 a(13) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 11 + 13 = 63.
%p A177176 seq(round(n*(n+1)*(2*n+1)/78),n=0..50)
%o A177176 (PARI) s=0;vector(90,n,s+=n^2\13)
%o A177176 (Magma) [Round(n*(n+1)*(2*n+1)/78): n in [0..60]]; // _Vincenzo Librandi_, Jun 23 2011
%Y A177176 Cf. A177100, A177116, A181120.
%K A177176 nonn,easy
%O A177176 0,5
%A A177176 _Mircea Merca_, Dec 10 2010