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%I #55 Oct 13 2020 03:26:36
%S 0,0,0,1,2,4,7,11,16,23,31,41,53,67,83,102,123,147,174,204,237,274,
%T 314,358,406,458,514,575,640,710,785,865,950,1041,1137,1239,1347,1461,
%U 1581,1708,1841,1981,2128,2282,2443,2612,2788,2972,3164,3364,3572
%N Partial sums of round(n^2/12) (A069905).
%C Number of triples of positive integers (a, b, c) such that 1 <= a <= b <= c and a + b + c <= n. - _Leonhard Vogt_, Apr 27 2017
%H Vincenzo Librandi, <a href="/A181120/b181120.txt">Table of n, a(n) for n = 0..870</a>
%H D. Barrera, M. J. Ibáñez, and S. Remogna, <a href="https://doi.org/10.1016/j.cam.2016.07.031">On the construction of trivariate near-best quasi-interpolants based on C^2 quartic splines on type-6 tetrahedral partitions</a>, Journal of Computational and Applied, 2016, Volume 311, February 2017, Pages 252-261.
%H J. Brandts and A. Cihangir, <a href="http://www.math.cas.cz/~am2013/proceedings/contributions/brandts.pdf">Counting triangles that share their vertices with the unit n-cube</a>, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
%H Jan Brandts and Apo Cihangir, <a href="http://arxiv.org/abs/1512.03044">Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group</a>, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
%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.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1).
%F a(n) = round((2*n^3 + 3*n^2 - 6*n)/72).
%F a(n) = round((4*n^3 + 6*n^2 - 12*n - 7)/144).
%F a(n) = floor((2*n^3 + 3*n^2 - 6*n + 9)/72).
%F a(n) = ceiling((2*n^3 + 3*n^2 - 6*n + 9 - 16)/72).
%F a(n) = a(n-6) + (n^2 - 5*n + 8)/2, n > 5.
%F From _R. J. Mathar_, Oct 06 2010: (Start)
%F a(n) = (-1)^n/16 + n^3/36 - n^2/24 - n/12 + 7/144 - A049347(n)/9.
%F G.f.: x^4 / ( (1+x)*(1+x+x^2)*(x-1)^4 ). (End)
%F a(n) = A000601(n-3). - _R. J. Mathar_, Oct 11 2017
%e a(5) = 4 = 0 + 0 + 0 + 1 + 1 + 2.
%p a:= n-> round(1/(72)*(2*n^(3)+3*n^(2)-6*n)): seq(a(n), n=0..50);
%o (PARI) a(n)=round(n*(2*n^2+3*n-6)/72) \\ _Charles R Greathouse IV_, May 23 2013
%Y Partial sums of A069905.
%K nonn,easy
%O 0,5
%A _Mircea Merca_, Oct 04 2010
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