|
%I #39 Aug 13 2022 06:23:11
%S 0,0,0,1,3,5,7,9,12,16,20,24,28,33,39,45,51,57,64,72,80,88,96,105,115,
%T 125,135,145,156,168,180,192,204,217,231,245,259,273,288,304,320,336,
%U 352,369,387,405,423,441,460,480,500,520,540,561,583,605,627,649,672
%N a(n) = floor(n^2/5).
%C It seems that for n >= 5, a(n) is the maximum number of non-overlapping 1 X 5 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's program. - _Dmitry Kamenetsky_, Aug 03 2009
%H Vincenzo Librandi, <a href="/A118015/b118015.txt">Table of n, a(n) for n = 0..5000</a>
%H R. D. Lobato, <a href="https://web.archive.org/web/20210506153254/http://lagrange.ime.usp.br/~lobato/packing/run/">Recursive partitioning approach for the Manufacturer's Pallet Loading Problem</a>.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).
%F G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^3). - _Klaus Brockhaus_, Nov 18 2008
%F a(n) = A008732(n-4) + A008732(n-3). - _R. J. Mathar_, Nov 22 2008
%F a(5*m+r) = m*(5*m + 2*r) + a(r), with m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*(5*4 + 2*3) + a(3) = 104 + 1 = 105. - _Bruno Berselli_, Dec 12 2016
%F Sum_{n>=3} 1/a(n) = 25/16 + Pi^2/30 + sqrt(5-2*sqrt(5))*Pi/4. - _Amiram Eldar_, Aug 13 2022
%t Table[Floor[n^2/5], {n, 0, 60}] (* _Bruno Berselli_, Dec 12 2016 *)
%o (Magma) [ n^2 div 5: n in [0..58] ]; // _Klaus Brockhaus_, Nov 18 2008
%o (PARI) a(n)=n^2\5 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Python) [int(n**2/5) for n in range(60)] # _Bruno Berselli_, Dec 12 2016
%o (Sage) [floor(n^2/5) for n in range(60)] # _Bruno Berselli_, Dec 12 2016
%Y Cf. A000212, A000290, A007590, A002620, A056827, A118013, A279169.
%Y Cf. A056834, A130519, A056838, A056865.
%K nonn,easy
%O 0,5
%A _Reinhard Zumkeller_, Apr 10 2006
%E Edited by _Charles R Greathouse IV_, Apr 20 2010
|
