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A177339 Partial sums of round(n^2/44).

%I A177339 #19 Sep 08 2022 08:45:53
%S A177339 0,0,0,0,0,1,2,3,4,6,8,11,14,18,22,27,33,40,47,55,64,74,85,97,110,124,
%T A177339 139,156,174,193,213,235,258,283,309,337,366,397,430,465,501,539,579,
%U A177339 621,665,711,759,809,861,916,973
%N A177339 Partial sums of round(n^2/44).
%C A177339 The round function is defined here by round(x) = floor(x + 1/2).
%C A177339 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 A177339 Vincenzo Librandi, <a href="/A177339/b177339.txt">Table of n, a(n) for n = 0..10000</a>
%H A177339 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 A177339 a(n) = round((2*n+1)*(2*n^2 + 2*n - 15)/528).
%F A177339 a(n) = floor((n+5)*(2*n^2 - 7*n + 21)/264).
%F A177339 a(n) = ceiling((n-4)*(2*n^2 + 11*n + 30)/264).
%F A177339 a(n) = round(n*(n-2)*(2*n+7)/264).
%F A177339 a(n) = a(n-44) + (n+1)*(n-44) + 665, n > 43.
%F A177339 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-11) - 2*a(n-12) + 2*a(n-14) - a(n-15) with g.f. x^5*(1 - x^2 + x^4) / ( (1+x) *(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 A177339 a(15) = 0 + 0 + 0 + 0 + 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + 5 = 27.
%p A177339 seq(round(n*(n-2)*(2*n+7)/264),n=0..50)
%o A177339 (Magma) [Round((2*n+1)*(2*n^2+2*n-15)/528): n in [0..60]]; // _Vincenzo Librandi_, Jun 23 2011
%Y A177339 Cf. A177100, A177116.
%K A177339 nonn,easy
%O A177339 0,7
%A A177339 _Mircea Merca_, Dec 10 2010