This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177116 #44 Oct 14 2024 00:10:50
%S A177116 0,0,0,1,2,4,7,11,17,24,33,44,57,72,90,110,133,159,188,221,257,297,
%T A177116 341,389,441,498,559,625,696,772,854,941,1034,1133,1238,1349,1467,
%U A177116 1591,1722,1860,2005,2158,2318,2486,2662,2846,3038,3239,3448,3666,3893
%N A177116 Partial sums of round(n^2/11).
%C A177116 The round function, also called the nearest integer function, is defined here by round(x)=floor(x+1/2).
%C A177116 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 A177116 Vincenzo Librandi, <a href="/A177116/b177116.txt">Table of n, a(n) for n = 0..870</a>
%H A177116 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 A177116 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1).
%F A177116 a(n) = round((n-2)*(n+3)*(2*n+1)/66).
%F A177116 a(n) = floor((2*n^3 + 3*n^2 - 11*n + 18)/66).
%F A177116 a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 30)/66).
%F A177116 a(n) = round(n*(2*n^2 + 3*n - 11)/66).
%F A177116 a(n) = a(n-11) + (n+1)*(n-11) + 44, n > 10.
%F A177116 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-11) - 3*a(n-12) + 3*a(n-13) - a(n-14). - _R. J. Mathar_, Dec 10 2010
%F A177116 G.f.: x^3 *(1+x) *(x^2-x+1) *(x^4-x^3+x^2-x+1) / ( (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 10 2010 [Typo fixed by _Colin Barker_, Oct 10 2012]
%e A177116 a(11) = 0 + 0 + 0 + 1 + 1 + 2 + 3 + 4 + 6 + 7 + 9 + 11 = 44.
%p A177116 seq(round((2*n^3+3*n^2-11*n)/66),n=0..50)
%t A177116 Accumulate[Round[Range[0,50]^2/11]] (* or *) LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,1,-3,3,-1},{0,0,0,1,2,4,7,11,17,24,33,44,57,72},60] (* _Harvey P. Dale_, Dec 10 2014 *)
%o A177116 (PARI) a(n)=(2*n^3+3*n^2-11*n+18)\66 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A177116 Cf. A173690 (k=5), A173691 (k=6), A173722 (k=8), A177100 (k=9), A181120 (k=12).
%K A177116 nonn,easy
%O A177116 0,5
%A A177116 _Mircea Merca_, Dec 09 2010