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a(n) = floor(n/2) * floor(n/3).
8

%I #64 Mar 30 2023 02:27:33

%S 0,0,1,2,2,6,6,8,12,15,15,24,24,28,35,40,40,54,54,60,70,77,77,96,96,

%T 104,117,126,126,150,150,160,176,187,187,216,216,228,247,260,260,294,

%U 294,308,330,345,345,384,384,400,425,442,442,486,486,504,532,551

%N a(n) = floor(n/2) * floor(n/3).

%C a(n) is also the number of 5 boxes polyomino (invert U patterns) packing into n X n square. The 6 boxes 2 X 3 (rectangular patterns) also gives the same sequence but difference in squares left. See illustration in links. - _Kival Ngaokrajang_, Nov 10 2013

%H Vincenzo Librandi, <a href="/A010762/b010762.txt">Table of n, a(n) for n = 1..1000</a>

%H Kival Ngaokrajang, <a href="/A010762/a010762_1.pdf">Illustration of initial terms of invert u and 2 X 3 rectangular patterns</a>.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,-1,1,0,-1,-1,0,1).

%F a(n) = A004526(n) * A002264(n). - _Reinhard Zumkeller_, Jul 25 2005

%F a(n) = a(n-2) + a(n-3) - a(n-5) + a(n-6) - a(n-8) - a(n-9) + a(n-11). - _Clark Kimberling_, May 18 2012

%F G.f.: -x^3*(x^7+x^6+x^5+2*x^4+3*x^3+x^2+2*x+1) / ((x-1)^3*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - _Colin Barker_, Apr 05 2013

%F Sum_{n>=3} (-1)^(n+1)/a(n) = sqrt(3)*Pi/4 + 9*log(3)/4 - 2*log(2) - 3/2. - _Amiram Eldar_, Mar 30 2023

%p [ seq(floor(n/2)*floor(n/3), n=1..64) ];

%t Table[Floor[n/2]*Floor[n/3], {n, 1, 70}] (* _Clark Kimberling_, May 18 2012 *)

%t CoefficientList[Series[- x^2 x^7 + x^6 + x^5 + 2 x^4 + 3 x^3 + x^2 + 2 x+1)/((x - 1)^3 (x + 1)^2 (x^2 - x + 1) (x^2 + x + 1)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 15 2013 *)

%t LinearRecurrence[{0,1,1,0,-1,1,0,-1,-1,0,1},{0,0,1,2,2,6,6,8,12,15,15},60] (* _Harvey P. Dale_, Jan 09 2016 *)

%o (Magma) [Floor(n/2)*Floor(n/3) : n in [1..50]]; // _Wesley Ivan Hurt_, Jun 22 2014

%o (PARI) a(n)=n\2 + n\3 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A002264, A004526, A010761, A110533, A242669.

%K nonn,easy

%O 1,4

%A _N. J. A. Sloane_, _Simon Plouffe_