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%I #22 Sep 08 2022 08:45:53
%S 0,0,0,0,0,1,2,3,5,7,10,13,17,22,27,33,40,48,57,67,78,90,103,118,134,
%T 151,170,190,212,235,260,287,315,345,377,411,447,485,525,567,611,658,
%U 707,758,812,868,927,988,1052,1119,1188
%N Partial sums of round(n^2/36).
%C The round function is defined here by round(x) = floor(x + 1/2).
%H Vincenzo Librandi, <a href="/A177337/b177337.txt">Table of n, a(n) for n = 0..910</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((2*n+1)*(2*n^2 + 2*n - 19)/432).
%F a(n) = floor((n+5)*(2*n^2 - 7*n + 17)/216).
%F a(n) = ceiling((n-4)*(2*n^2 + 11*n + 26)/216).
%F a(n) = round((2*n^3 + 3*n^2 - 18*n)/216).
%F a(n) = a(n-36) + (n+1)*(n-36) + 447, n > 35.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-18) - 3*a(n-19) + 3*a(n-20) - a(n-21) with g.f. x^5*(1 - x + x^3 - x^4 + x^5 - x^6 + x^7 - x^9 + x^10) / ( (1+x) *(1 + x + x^2) *(x^2 - x + 1) *(x^6 + x^3 + 1) *(x^6 - x^3 + 1) *(x-1)^4 ). - _R. J. Mathar_, Dec 13 2010
%e a(19) = 0 + 0 + 0 + 0 + 0 + 1 + 1 + 1 + 2 + 2 + 3 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 9 + 10 = 67.
%p seq(round((2*n^3+3*n^2-18*n)/216),n=0..50)
%t Accumulate[Floor[Range[0,60]^2/36+1/2]] (* _Harvey P. Dale_, Sep 29 2011 *)
%o (Magma) [Floor((n+5)*(2*n^2-7*n+17)/216): n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011
%Y Cf. A177100, A177116.
%K nonn,easy
%O 0,7
%A _Mircea Merca_, Dec 10 2010
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