cp's OEIS Frontend

A181286 Partial sums of floor(n^2/3) (A000212).

%I A181286 #40 Oct 24 2022 11:09:46
%S A181286 0,0,1,4,9,17,29,45,66,93,126,166,214,270,335,410,495,591,699,819,952,
%T A181286 1099,1260,1436,1628,1836,2061,2304,2565,2845,3145,3465,3806,4169,
%U A181286 4554,4962,5394,5850,6331,6838,7371,7931,8519,9135,9780,10455,11160,11896
%N A181286 Partial sums of floor(n^2/3) (A000212).
%C A181286 Column sums of:
%C A181286 1 4 9 16 25 36 49...
%C A181286        1  4  9 16...
%C A181286                 1...
%C A181286 ....................
%C A181286 --------------------
%C A181286 1 4 9 17 29 45 66...
%H A181286 Vincenzo Librandi, <a href="/A181286/b181286.txt">Table of n, a(n) for n = 0..5000</a>
%H A181286 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 A181286 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A181286 a(n) = Sum_{k=0..n} floor(k^2/3).
%F A181286 a(n) = round((2*n^3 + 3*n^2 - 3*n)/18).
%F A181286 a(n) = round((2*n^3 + 3*n^2 - 3*n - 2)/18).
%F A181286 a(n) = floor((2*n^3 + 3*n^2 - 3*n)/18).
%F A181286 a(n) = ceiling((2*n^3 + 3*n^2 - 3*n - 4)/18).
%F A181286 a(n) = a(n-3) + (n-1)^2, n > 2.
%F A181286 G.f.: x^2*(1+x)/((1-x)^3*(1-x^3)).
%F A181286 G.f.: x^2*(1+x)/((1+x+x^2)*(1-x)^4). - _L. Edson Jeffery_, Jan 16 2014
%F A181286 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6), n >= 6. - _L. Edson Jeffery_, Jan 16 2014
%F A181286 E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(-2 + x*(2 + x*(9 + 2*x))) + 6*cos(sqrt(3)*x/2) + 2*sqrt(3)*sin(sqrt(3)*x/2))/54. - _Stefano Spezia_, Oct 24 2022
%e A181286 a(5) = 17 = 0 + 0 + 1 + 3 + 5 + 8.
%p A181286 a(n):=round((2*n^(3)+3*n^(2)-3*n)/(18))
%t A181286 Accumulate[Floor[Range[0,80]^2/3]] (* _Harvey P. Dale_, Jun 14 2015 *)
%o A181286 (Magma) [Round((2*n^(3)+3*n^(2)-3*n)/(18)): n in [0..50]]; // _Vincenzo Librandi_, Jun 21 2011
%Y A181286 Cf. A000212.
%K A181286 nonn,easy
%O A181286 0,4
%A A181286 _Mircea Merca_, Oct 12 2010