%I #22 Sep 08 2022 08:45:53

%S 0,0,0,1,2,3,5,8,12,17,23,30,38,48,60,73,88,105,124,145,169,195,223,

%T 254,288,325,365,408,454,503,556,613,673,737,805,877,953,1034,1119,

%U 1208,1302,1401,1505,1614,1728,1847,1971,2101,2237,2378,2525

%N Partial sums of round(n^2/17).

%C The round 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="/A177205/b177205.txt">Table of n, a(n) for n = 0..890</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(n*(n+1)*(2*n+1)/102).

%F a(n) = floor((2*n^3 + 3*n^2 + n + 36)/102).

%F a(n) = ceiling((2*n^3 + 3*n^2 + n - 36)/102).

%F a(n) = a(n-17) + (n+1)*(n-17) + 105, n > 16.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-17) - 3*a(n-18) + 3*a(n-19) - a(n-20) with g.f. x^3 *(1+x) *(x^12 - 2*x^11 + 2*x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + 2*x^2 - 2*x + 1) / ( (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) *(x-1)^4 ). - _R. J. Mathar_, Dec 13 2010

%e a(17) = 0 + 0 + 0 + 1 + 1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 12 + 13 + 15 + 17 = 105.

%p seq(round(n*(n+1)*(2*n+1)/102),n=0..50)

%t Accumulate[Round[Range[0,50]^2/17]] (* _Harvey P. Dale_, Jul 04 2022 *)

%o (Magma) [Floor((2*n^3+3*n^2+n+36)/102): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011

%Y Cf. A177100, A177116.

%K nonn,easy

%O 0,5

%A _Mircea Merca_, Dec 10 2010