Search: id:A000212
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%I A000212 M2439 N0966
%S A000212 0,0,1,3,5,8,12,16,21,27,33,40,48,56,65,75,85,96,108,120,133,147,
%T A000212 161,176,192,208,225,243,261,280,300,320,341,363,385,408,432,456,
%U A000212 481,507,533,560,588,616,645,675,705,736,768,800,833,867,901,936
%N A000212 [n^2/3].
%C A000212 Let M_n be the n X n matrix of the following form [3 2 1 0 0 0 0 0 0 
               0 / 2 3 2 1 0 0 0 0 0 0 / 1 2 3 2 1 0 0 0 0 0 / 0 1 2 3 2 1 0 0 0 
               0 / 0 0 1 2 3 2 1 0 0 0 / 0 0 0 1 2 3 2 1 0 0 / 0 0 0 0 1 2 3 2 1 
               0 / 0 0 0 0 0 1 2 3 2 1 / 0 0 0 0 0 0 1 2 3 2 / 0 0 0 0 0 0 0 1 2 
               3]. Then for n>2 a(n) = det M_(n-2). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 20 2002
%C A000212 Largest possible size for the directed Cayley graph on two generators 
               having diameter n-2. - Ralf Stephan, Apr 27 2003
%C A000212 It seems that for n >= 2 a(n) = maximum number of non-overlapping 1x3 
               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 
               http://lagrange.ime.usp.br/~lobato/packing/run/index.php [From Dmitry 
               Kamenetsky (dkamen(AT)rsise.anu.edu.au), Aug 03 2009]
%D A000212 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000212 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000212 C. K. Wong and D. Coppersmith, A combinatorial problem related to multimodule 
               memory organizations, J. ACM 21 (1974), 392-402.
%H A000212 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000212 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A000212 G.f.: x^2*(1+x)/((1-x)^2*(1-x^3)) - Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Apr 01 2002
%F A000212 Euler transform of length 3 sequence [ 3, -1, 1]. - Michael Somos Sep 
               25 2006
%F A000212 G.f.: x^2*(1-x^2)/((1-x)^3*(1-x^3)). a(-n)=a(n). - Michael Somos Sep 
               25 2006
%p A000212 A000212:=(-1+z-2*z**2+z**3-2*z**4+z**5)/(z**2+z+1)/(z-1)**3; [Conjectured 
               by S. Plouffe in his 1992 dissertation. Gives sequence with an additional 
               leading 1.]
%t A000212 k0=k1=0;lst={k0, k1};Do[kt=k1;k1=n^2-k1-k0;k0=kt;AppendTo[lst, k1], {n, 
               1, 5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 
               11 2008]
%o A000212 (PARI) a(n)=n^2\3
%Y A000212 Cf. A000290, A007590, A002620, A118015, A056827, A118013.
%Y A000212 Sequence in context: A023660 A161339 A023562 this_sequence A094913 A020678 
               A014811
%Y A000212 Adjacent sequences: A000209 A000210 A000211 this_sequence A000213 A000214 
               A000215
%K A000212 nonn
%O A000212 0,4
%A A000212 N. J. A. Sloane (njas(AT)research.att.com).

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