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%I #57 Sep 08 2022 08:45:31
%S 0,1,3,8,16,29,47,72,104,145,195,256,328,413,511,624,752,897,1059,
%T 1240,1440,1661,1903,2168,2456,2769,3107,3472,3864,4285,4735,5216,
%U 5728,6273,6851,7464,8112,8797,9519,10280,11080,11921,12803,13728,14696,15709
%N Partial sums of ceiling(n^2/2) (A000982).
%C Binomial transform of [0, 1, 1, 2, -2, 4, -8, 16, -32, ...].
%C Starting with offset 1 = (1, 3, 5, 7, ...) convolved with (1, 0, 3, 0, 5, ...). - _Gary W. Adamson_, Feb 16 2009
%H Vincenzo Librandi, <a href="/A131941/b131941.txt">Table of n, a(n) for n = 0..10000</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, p. 11.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F For even n, a(n) = n*(2*n^2 +3*n +4)/12. For odd n, a(n) = (n+1)*(2*n^2 +n +3)/12. - _Washington Bomfim_, Jul 31 2008
%F From _R. J. Mathar_, Feb 24 2010: (Start)
%F G.f.: x*(1+x^2)/((1+x)*(1-x)^4).
%F a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)
%F From _Mircea Merca_, Oct 10 2010: (Start)
%F a(n) = round((2*n^3 + 3*n^2 + 4*n)/12) = round((2*n+1)*(2*n^2 + 3*n + 3)/24) = floor((n+1)*(2*n^2 + n + 3)/12) = ceiling((2*n^3 + 3*n^2 + 4*n)/12).
%F a(n) = a(n-2) + n^2 - n + 1, n > 1. (End)
%F a(n) = (2*n*(2*n^2 + 3*n + 4) - 3*(-1)^n + 3)/24. - _Bruno Berselli_, Dec 07 2010
%F E.g.f.: (x*(9 + 9*x + 2*x^2)*cosh(x) + (3 + 9*x + 9*x^2 + 2*x^3)*sinh(x))/12. - _Stefano Spezia_, Dec 21 2021
%e a(3) = 8 = 0 + 1 + 2 + 5.
%p a(n):=round(1/(12)(2*n^(3)+3*n^(2)+4*n)) # _Mircea Merca_, Oct 10 2010
%t CoefficientList[Series[x (1 + x^2)/(1 + x)/(1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)
%o (PARI) a(n) = (n+[0,1][n%2+1]) * (2*n^2 +[3,1][n%2+1]*n +[4,3][n%2+1])/12 \\ _Washington Bomfim_, Jul 31 2008
%o (Magma) [Ceiling((2*n^3+3*n^2+4*n)/12): n in [0..60]]; // _Vincenzo Librandi_, Jun 25 2011
%Y Cf. A000982, A080930 (binomial transform without leading 0).
%K nonn,easy
%O 0,3
%A _Gary W. Adamson_, Oct 25 2007
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