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A172046 Partial sums of floor(n^2/7) (A056834).

%I A172046 #33 Sep 08 2022 08:45:50
%S A172046 0,0,0,1,3,6,11,18,27,38,52,69,89,113,141,173,209,250,296,347,404,467,
%T A172046 536,611,693,782,878,982,1094,1214,1342,1479,1625,1780,1945,2120,2305,
%U A172046 2500,2706,2923,3151,3391,3643,3907,4183,4472,4774,5089,5418,5761,6118
%N A172046 Partial sums of floor(n^2/7) (A056834).
%H A172046 Vincenzo Librandi, <a href="/A172046/b172046.txt">Table of n, a(n) for n = 0..10000</a>
%H A172046 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 A172046 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,0,1,-3,3,-1).
%F A172046 a(n) = Sum_{k=0..n} floor(k^2/7).
%F A172046 a(n) = round((2*n^3 + 3*n^2 - 11*n)/42).
%F A172046 a(n) = round((2*n^3 + 3*n^2 - 11*n - 6)/42).
%F A172046 a(n) = floor((2*n^3 + 3*n^2 - 11*n + 6)/42).
%F A172046 a(n) = ceiling((2*n^3 + 3*n^2 - 11*n - 18)/42).
%F A172046 a(n) = a(n-7) + (n-2)*(n-4) + 3, n > 6.
%F A172046 G.f.: x^3*(x+1)*(x^2 - x + 1)/((x-1)^4*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). [_Colin Barker_, Oct 26 2012]
%e A172046 a(5) = 6 = 0 + 0 + 0 + 1 + 2 + 3.
%p A172046 a:= n-> round((2*n^3+3*n^2-11*n)/42): seq (a(n), n=0..50);
%t A172046 LinearRecurrence[{3,-3,1,0,0,0,1,-3,3,-1},{0,0,0,1,3,6,11,18,27,38},60] (* _Harvey P. Dale_, Apr 12 2017 *)
%o A172046 (Magma) [Round((2*n^3+3*n^2-11*n)/42): n in [0..60]]; // _Vincenzo Librandi_, Jun 25 2011
%Y A172046 Cf. A056834.
%K A172046 nonn,easy
%O A172046 0,5
%A A172046 _Mircea Merca_, Nov 19 2010