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A000212
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Floor(n^2/3).
(Formerly M2439 N0966)
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31
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0, 0, 1, 3, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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, Jun 20 2002
Largest possible size for the directed Cayley graph on two generators having diameter n-2. - Ralf Stephan, Apr 27 2003
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 Lobato's tool, see links. [From Dmitry Kamenetsky, Aug 03 2009]
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REFERENCES
| N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
C. K. Wong and D. Coppersmith, A combinatorial problem related to multimodule memory organizations, J. ACM 21 (1974), 392-402.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..5000
R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (2,-1,1,-2,1)
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FORMULA
| G.f.: x^2*(1+x)/((1-x)^2*(1-x^3)) - Frank Adams-Watters, Apr 01 2002
Euler transform of length 3 sequence [ 3, -1, 1]. - Michael Somos Sep 25 2006
G.f.: x^2*(1-x^2)/((1-x)^3*(1-x^3)). a(-n)=a(n). - Michael Somos Sep 25 2006
a(n) = sum(A011655(k)*(n-k): 0<=k<=n). [From Reinhard Zumkeller,Nov 30 2009]
a(n)=a(n-1)+a(n-3)-a(n-4)+2 for n>=4. - Alexander Burstein, Nov 20 2011
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MAPLE
| 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.]
A000212 := proc(n) option remember; `if`(n<4, [0, 0, 1, 3][n+1], a(n-1)+a(n-3)-a(n-4)+2) end; # Peter Luschny, Nov 20 2011
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MATHEMATICA
| 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, Dec 11 2008]
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PROG
| (PARI) a(n)=n^2\3
(MAGMA) [Floor(n^2 / 3): n in [0..50]]; // Vincenzo Librandi, May 08 2011
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CROSSREFS
| Cf. A000290, A007590, A002620, A118015, A056827, A118013.
Cf. A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033443, A033444. [From Reinhard Zumkeller, Nov 30 2009]
Sequence in context: A194176 A186494 A194180 * A183139 A094913 A020678
Adjacent sequences: A000209 A000210 A000211 * A000213 A000214 A000215
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Charles R Greathouse IV, Apr 19 2010
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