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A006999
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Partitioning integers to avoid arithmetic progressions of length 3.
(Formerly M1047)
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6
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0, 1, 2, 4, 7, 11, 17, 26, 40, 61, 92, 139, 209, 314, 472, 709, 1064, 1597, 2396, 3595, 5393, 8090, 12136, 18205, 27308, 40963, 61445, 92168, 138253, 207380, 311071, 466607, 699911, 1049867, 1574801, 2362202, 3543304, 5314957, 7972436
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = A006997(3^n-1).
It appears that, aside from the first term, this is the (L)-sieve transform of A016789 ={2,5,8,11,...,3n+2....}. This has been verified up to a(30)=311071. See A152009 for the definition of the (L)-sieve transform. [From John W. Layman (layman(AT)math.vt.edu), Nov 20 2008]
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REFERENCES
| Gerver, Joseph; Propp, James; Simpson, Jamie; Greedily partitioning the natural numbers into sets free of arithmetic progressions. Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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FORMULA
| a(n) = [ (3a(n-1)+2)/2 ].
a(n)=-1+floor(c*(3/2)^n) where c=1.081513668589844877304633988599549408710737041542024954790295591585622666484989650922411026555488940... - Benoit Cloitre, Jan 10, 2002
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PROG
| (PARI) a(n)=if(n<1, 0, floor((3*a(n-1)+2)/2))
(Haskell)
a006999 n = a006999_list !! n
a006999_list = 0 : map ((`div` 2) . (+ 2) . (* 3)) a006999_list
-- Reinhard Zumkeller, Oct 26 2011
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CROSSREFS
| a(n) = A061419(n) - 1 = A061418(n) - 2.
The constant c is 2/3*K(3) (see A083286). - Ralf Stephan, May 29, 2003
Cf. A003312.
First differences are in A073941.
Cf. A016789, A152009 [From John W. Layman (layman(AT)math.vt.edu), Nov 20 2008]
Cf. A005428 (first differences).
Sequence in context: A096967 A117276 A035295 * A005252 A023430 A023429
Adjacent sequences: A006996 A006997 A006998 * A007000 A007001 A007002
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), D.R. Hofstadter, Jim Propp (propp(AT)math.wisc.edu)
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 06 2000
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