cp's OEIS Frontend

A173690 Partial sums of round(n^2/5).

%I A173690 #42 Oct 12 2024 03:17:21
%S A173690 0,0,1,3,6,11,18,28,41,57,77,101,130,164,203,248,299,357,422,494,574,
%T A173690 662,759,865,980,1105,1240,1386,1543,1711,1891,2083,2288,2506,2737,
%U A173690 2982,3241,3515,3804,4108,4428,4764,5117,5487,5874,6279,6702,7144,7605,8085,8585
%N A173690 Partial sums of round(n^2/5).
%C A173690 Partial sums of A008738.
%H A173690 Vincenzo Librandi, <a href="/A173690/b173690.txt">Table of n, a(n) for n = 0..2000</a>
%H A173690 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 A173690 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1,-3,3,-1).
%F A173690 a(n) = Sum_{k=0..n} round(k^2/5);
%F A173690 a(n) = round((2*n^3 + 3*n^2 + n)/30);
%F A173690 a(n) = floor((2*n^3 + 3*n^2 + n + 6)/30);
%F A173690 a(n) = ceiling((2*n^3 + 3*n^2 + n - 6)/30);
%F A173690 a(n) = a(n-5) + (n-2)^2 + 2, n > 4;
%F A173690 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-5) - 3*a(n-6) + 3*a(n-7) - a(n-8), n > 7.
%F A173690 G.f.: x^2*(x+1)*(x^2 - x + 1) / ( (x^4 + x^3 + x^2 + x + 1)*(x-1)^4 ).
%e A173690 a(5) = round(1/5) + round(4/5) + round(9/5) + round(16/5) + round(25/5) = 0 + 1 + 2 + 3 + 5 = 11.
%p A173690 A173690 := proc(n) add( round(i^2/5),i=0..n) ; end proc: # _R. J. Mathar_, Jan 10 2011
%t A173690 Accumulate[Round[Range[0,50]^2/5]] (* or *) LinearRecurrence[{3,-3,1,0,1,-3,3,-1},{0,0,1,3,6,11,18,28},60] (* _Harvey P. Dale_, Mar 16 2022 *)
%o A173690 (PARI) a(n)=(2*n^3+3*n^2+n+6)\30 \\ _Charles R Greathouse IV_, May 30 2011
%Y A173690 Cf. A008738.
%K A173690 nonn,easy
%O A173690 0,4
%A A173690 _Mircea Merca_, Nov 25 2010