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

%I #42 Oct 12 2024 03:17:21

%S 0,0,1,3,6,11,18,28,41,57,77,101,130,164,203,248,299,357,422,494,574,

%T 662,759,865,980,1105,1240,1386,1543,1711,1891,2083,2288,2506,2737,

%U 2982,3241,3515,3804,4108,4428,4764,5117,5487,5874,6279,6702,7144,7605,8085,8585

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

%C Partial sums of A008738.

%H Vincenzo Librandi, <a href="/A173690/b173690.txt">Table of n, a(n) for n = 0..2000</a>

%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_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,1,-3,3,-1).

%F a(n) = Sum_{k=0..n} round(k^2/5);

%F a(n) = round((2*n^3 + 3*n^2 + n)/30);

%F a(n) = floor((2*n^3 + 3*n^2 + n + 6)/30);

%F a(n) = ceiling((2*n^3 + 3*n^2 + n - 6)/30);

%F a(n) = a(n-5) + (n-2)^2 + 2, n > 4;

%F 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 G.f.: x^2*(x+1)*(x^2 - x + 1) / ( (x^4 + x^3 + x^2 + x + 1)*(x-1)^4 ).

%e 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 := proc(n) add( round(i^2/5),i=0..n) ; end proc: # _R. J. Mathar_, Jan 10 2011

%t 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 (PARI) a(n)=(2*n^3+3*n^2+n+6)\30 \\ _Charles R Greathouse IV_, May 30 2011

%Y Cf. A008738.

%K nonn,easy

%O 0,4

%A _Mircea Merca_, Nov 25 2010