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%I #19 Feb 20 2024 03:25:01
%S 0,0,0,1,2,2,5,5,7,11,14,14,20,20,24,31,36,36,45,45,51,61,68,68,80,80,
%T 88,101,110,110,125,125,135,151,162,162,180,180,192,211,224,224,245,
%U 245,259,281,296,296,320,320,336,361,378,378,405,405,423,451
%N a(n) = floor(n/2)*floor(n/3)-floor(n/6)^2.
%H Vincenzo Librandi, <a href="/A213032/b213032.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,2,-2,0,0,0,0,-1,1).
%F a(n) = a(n-1)+2*a(n-6)-2*a(n-7)-a(n-12)+a(n-13).
%F G.f.: (x^3 + x^4 + 3*x^6 + 2*x^8 + 2*x^9 + x^10)/(1 - x - 2*x^6 + 2*x^7 + x^12 - x^13).
%t a[n_] := Floor[n/2]*Floor[n/3] - Floor[n/6]^2
%t Table[a[n], {n, 0, 90}] (* A213032 *)
%t LinearRecurrence[{1, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, -1, 1}, {0, 0, 0, 1, 2, 2, 5, 5, 7, 11, 14, 14, 20}, 90]
%t CoefficientList[Series[(x^3 + x^4 + 3 x^6 + 2 x^8 + 2 x^9 + x^10) / (1 - x - 2 x^6 + 2 x^7 + x^12 - x^13), {x, 0, 80}], x] (* _Vincenzo Librandi_, Aug 02 2013 *)
%o (Magma) [Floor(n/2)*Floor(n/3)-Floor(n/6)^2: n in [0..80]]; // _Vincenzo Librandi_, Aug 02 2013
%K nonn,easy
%O 0,5
%A _Clark Kimberling_, Jun 05 2012
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