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A172131 Partial sums of floor(n^2/9) (A056838).

%I A172131 #29 Sep 08 2022 08:45:50
%S A172131 0,0,0,1,2,4,8,13,20,29,40,53,69,87,108,133,161,193,229,269,313,362,
%T A172131 415,473,537,606,681,762,849,942,1042,1148,1261,1382,1510,1646,1790,
%U A172131 1942,2102,2271,2448,2634,2830,3035,3250,3475,3710,3955,4211,4477,4754
%N A172131 Partial sums of floor(n^2/9) (A056838).
%H A172131 Vincenzo Librandi, <a href="/A172131/b172131.txt">Table of n, a(n) for n = 0..10000</a>
%H A172131 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 A172131 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,0,0,1,-3,3,-1).
%F A172131 a(n) = Sum_{k=0..n} floor(k^2/9).
%F A172131 a(n) = round((2*n^3 + 3*n^2 - 15*n - 9)/54).
%F A172131 a(n) = round((2*n^3 + 3*n^2 - 15*n - 8)/54).
%F A172131 a(n) = floor((2*n^3 + 3*n^2 - 15*n + 18)/54).
%F A172131 a(n) = ceiling((2*n^3 + 3*n^2 - 15*n - 34)/54).
%F A172131 a(n) = a(n-9) + (n-4)^2 + 4, n > 8.
%F A172131 G.f.: x^3*(x+1)*(x^2 - x + 1)^2/((x-1)^4*(x^2 + x + 1)*(x^6 + x^3 + 1)). [_Colin Barker_, Oct 26 2012]
%e A172131 a(6) = 8 = 0 + 0 + 0 + 1 + 1 + 2 + 4.
%p A172131 a:= n-> round((2*n^3+3*n^2-15*n-9)/54): seq (a(n), n=0..50);
%t A172131 Accumulate[Floor[Range[0,50]^2/9]] (* or *) LinearRecurrence[{3,-3,1,0,0,0,0,0,1,-3,3,-1},{0,0,0,1,2,4,8,13,20,29,40,53},60] (* _Harvey P. Dale_, Jan 10 2020 *)
%o A172131 (Magma) [Round((2*n^3+3*n^2-15*n-9)/54): n in [0..60]]; // _Vincenzo Librandi_, Jun 25 2011
%Y A172131 Cf. A056838.
%K A172131 nonn,easy
%O A172131 0,5
%A A172131 _Mircea Merca_, Nov 19 2010