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A008130 a(n) = floor(n/3)*ceiling(n/3).

%I A008130 #41 May 10 2025 03:16:54
%S A008130 0,0,0,1,2,2,4,6,6,9,12,12,16,20,20,25,30,30,36,42,42,49,56,56,64,72,
%T A008130 72,81,90,90,100,110,110,121,132,132,144,156,156,169,182,182,196,210,
%U A008130 210,225,240,240,256,272,272,289,306,306,324,342,342,361,380,380,400
%N A008130 a(n) = floor(n/3)*ceiling(n/3).
%H A008130 Vincenzo Librandi, <a href="/A008130/b008130.txt">Table of n, a(n) for n = 0..1000</a>
%H A008130 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A008130 From _R. J. Mathar_, Jan 27 2011: (Start)
%F A008130 a(n) = A002264(n)*A002264(n+2).
%F A008130 G.f. -x^3*(1+x) / ( (1+x+x^2)^2*(x-1)^3 ). (End)
%F A008130 From _Amiram Eldar_, May 10 2025: (Start)
%F A008130 Sum_{n>=3} 1/a(n) = 2 + Pi^2/6.
%F A008130 Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). (End)
%p A008130 A008130:=n->ceil(n/3)*floor(n/3); seq(A008130(n), n=0..60); # _Wesley Ivan Hurt_, Feb 01 2014
%t A008130 f[n_]:=Ceiling[n/3]*Floor[n/3];Table[f[n],{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 01 2010 *)
%t A008130 CoefficientList[Series[- x^3 (1 + x) / ((1 + x + x^2)^2 (x - 1)^3), {x, 0, 60}], x] (* _Vincenzo Librandi_, Jun 10 2013 *)
%t A008130 LinearRecurrence[{1,0,2,-2,0,-1,1},{0,0,0,1,2,2,4},60] (* _Harvey P. Dale_, Dec 31 2016 *)
%o A008130 (Magma) [Floor(n/3)*Ceiling(n/3): n in [0..60]]; // _Vincenzo Librandi_, Jun 10 2013
%Y A008130 Cf. A002264, A072691.
%K A008130 nonn,easy
%O A008130 0,5
%A A008130 _N. J. A. Sloane_