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 A177332 #34 Feb 01 2023 02:30:25
%S A177332 0,0,0,0,1,2,3,5,7,10,13,17,22,28,35,43,52,62,73,85,99,114,131,149,
%T A177332 169,191,214,239,266,295,326,359,394,432,472,514,559,606,656,708,763,
%U A177332 821,882,946,1013,1083,1156,1232,1311,1394,1480
%N A177332 Partial sums of round(n^2/29).
%C A177332 The round function is defined here by round(x) = floor(x + 1/2).
%C A177332 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 A177332 Vincenzo Librandi, <a href="/A177332/b177332.txt">Table of n, a(n) for n = 0..905</a>
%H A177332 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 A177332 <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1).
%F A177332 a(n) = round(n*(n+1)*(2*n+1)/174).
%F A177332 a(n) = floor((n+4)*(2*n^2 - 5*n + 21)/174).
%F A177332 a(n) = ceiling((n-3)*(2*n^2 + 9*n + 28)/174).
%F A177332 a(n) = a(n-29) + (n+1)*(n-29) + 266, n > 28.
%F A177332 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-29) - 3*a(n-30) + 3*a(n-31) - a(n-32). - _R. J. Mathar_, Dec 13 2010
%F A177332 G.f.: x^4*(x+1)*(x^2 - x + 1)*(x^4 - x^2 + 1)*(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)*(x^10 - x^6 + x^5 - x^4 + 1)/((x-1)^4*(x^28 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^20 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). - _Colin Barker_, Apr 06 2012
%e A177332 a(17) = 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 62.
%p A177332 seq(round(n*(n+1)*(2*n+1)/174),n=0..50)
%t A177332 Accumulate[Table[Round[n^2/29],{n,0,60}]] (* _Harvey P. Dale_, Dec 18 2010 *)
%o A177332 (Magma) [Floor((n+4)*(2*n^2-5*n+21)/174): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011
%o A177332 (PARI) a(n)=(2*n^3+3*n^2+n+84)\174 \\ _Charles R Greathouse IV_, Apr 06 2012
%o A177332 (Python)
%o A177332 def A177332(n): return (n*(n*(2*n + 3) + 1) + 84)//174 # _Chai Wah Wu_, Jan 31 2023
%Y A177332 Cf. A177100, A177116.
%K A177332 nonn,easy
%O A177332 0,6
%A A177332 _Mircea Merca_, Dec 10 2010