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%I #34 Oct 19 2022 08:17:31
%S 0,0,1,2,4,7,12,18,26,36,49,64,82,103,128,156,188,224,265,310,360,415,
%T 476,542,614,692,777,868,966,1071,1184,1304,1432,1568,1713,1866,2028,
%U 2199,2380,2570,2770,2980,3201,3432,3674,3927,4192,4468,4756,5056,5369
%N Partial sums of round(n^2/8).
%C Partial sums of A001971.
%H Vincenzo Librandi, <a href="/A173722/b173722.txt">Table of n, a(n) for n = 0..5000</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_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-3,3,-1).
%F a(n) = Sum_{k=0..n} round(k^2/8).
%F a(n) = round((2*n^3+3*n^2+4*n)/48).
%F a(n) = round((2*n+1)*(2*n^2+2*n+3)/96).
%F a(n) = floor((n+2)*(2*n^2-n+6)/48).
%F a(n) = ceiling((2*n^3+3*n^2+4*n-9)/48).
%F a(n) = a(n-4)+n*(n-3)/2+2, n>3.
%F G.f.: x^2*(1-x+x^2) / ( (1+x)*(x^2+1)*(x-1)^4 ). - _R. J. Mathar_, Nov 26 2010
%F a(n) = 3*(-1)^n/32+n^2/16+n/12+n^3/24+1/32-A057077(n)/8. - _R. J. Mathar_, Nov 26 2010
%e a(5) = round(1/8) + round(4/8) + round(9/8) + round(16/8) + round(25/8) = 0 + 1 + 1 + 2 + 3 = 7.
%p A057077 := proc(n) op( 1+(n mod 4),[1,1,-1,-1]) ; end proc:
%p A173722 := proc(n) 3*(-1)^n/32+n^2/16+n/12+n^3/24+1/32-A057077(n)/8 ; end proc:
%p seq(A173722(n),n=0..80) ; # _R. J. Mathar_, Nov 26 2010
%t Table[Floor[(n + 2)*(2*n^2 - n + 6)/48], {n,0,50}] (* _G. C. Greubel_, Nov 29 2016 *)
%o (PARI) a(n)=(n+2)*(2*n^2-n+6)\48 \\ _Charles R Greathouse IV_, Oct 19 2022
%Y Cf. A001971.
%K nonn,easy
%O 0,4
%A _Mircea Merca_, Nov 26 2010
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