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A129371 a(n) = Sum_{k=0..floor(n/2)} (n-k)^2.

%I A129371 #19 Jan 31 2024 13:12:15
%S A129371 0,1,5,13,29,50,86,126,190,255,355,451,595,728,924,1100,1356,1581,
%T A129371 1905,2185,2585,2926,3410,3818,4394,4875,5551,6111,6895,7540,8440,
%U A129371 9176,10200,11033,12189,13125,14421,15466,16910,18070
%N A129371 a(n) = Sum_{k=0..floor(n/2)} (n-k)^2.
%C A129371 Partial sums of A129370.
%C A129371 Partial row sums of A055461. - _G. C. Greubel_, Jan 31 2024
%H A129371 G. C. Greubel, <a href="/A129371/b129371.txt">Table of n, a(n) for n = 0..5000</a>
%H A129371 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A129371 G.f.: x*(1+4*x+5*x^2+4*x^3)/((1-x)*(1-x^2)^3).
%F A129371 a(n) = Sum_{k = floor((n+1)/2)..n} k^2.
%F A129371 From _R. J. Mathar_, Apr 21 2010: (Start)
%F A129371 a(n) = a(n-1) +3*a(n-2) -3*a(n-3) -3*a(n-4) +3*a(n-5) +a(n-6) -a(n-7).
%F A129371 a(n) = 7*n^3/24 + 9*n^2/16 + 7*n/48 + (-1)^n*n*(n-1)/16. (End)
%t A129371 Accumulate[Table[n^2-(n-1)^2 (1-(-1)^n)/8,{n,0,50}]] (* _Harvey P. Dale_, Oct 22 2011 *)
%o A129371 (Magma) [n*(14*n^2+27*n+7)/48 +(-1)^n*Binomial(n,2)/8: n in [0..60]]; // _G. C. Greubel_, Jan 31 2024
%o A129371 (SageMath) [n*(14*n^2+27*n+7)/48 +(-1)^n*binomial(n,2)/8 for n in range(61)] # _G. C. Greubel_, Jan 31 2024
%Y A129371 Cf. A055461, A129370.
%K A129371 easy,nonn
%O A129371 0,3
%A A129371 _Paul Barry_, Apr 11 2007