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A006999 Partitioning integers to avoid arithmetic progressions of length 3.

%I A006999 M1047 #71 Aug 10 2025 20:42:21
%S A006999 0,1,2,4,7,11,17,26,40,61,92,139,209,314,472,709,1064,1597,2396,3595,
%T A006999 5393,8090,12136,18205,27308,40963,61445,92168,138253,207380,311071,
%U A006999 466607,699911,1049867,1574801,2362202,3543304,5314957,7972436
%N A006999 Partitioning integers to avoid arithmetic progressions of length 3.
%C A006999 a(n) = A006997(3^n-1).
%C A006999 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. - _John W. Layman_, Nov 20 2008
%C A006999 a(n) is also the largest-index square reachable in n jumps if we start at square 0 of the Infinite Sidewalk. - _Jose Villegas_, Mar 27 2023
%D A006999 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006999 Reinhard Zumkeller, <a href="/A006999/b006999.txt">Table of n, a(n) for n = 0..1000</a>
%H A006999 B. Chen, R. Chen, J. Guo, S. Lee et al, <a href="http://arxiv.org/abs/1808.04304">On Base 3/2 and its sequences</a>, arXiv:1808.04304 [math.NT], 2018.
%H A006999 Joseph Gerver, James Propp and Jamie Simpson, <a href="http://dx.doi.org/10.1090/S0002-9939-1988-0929018-1">Greedily partitioning the natural numbers into sets free of arithmetic progressions</a> Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772.
%H A006999 D. R. Hofstadter, <a href="/A006336/a006336_1.pdf">Eta-Lore</a> [Cached copy, with permission]
%H A006999 D. R. Hofstadter, <a href="/A006336/a006336_2.pdf">Pi-Mu Sequences</a> [Cached copy, with permission]
%H A006999 D. R. Hofstadter and N. J. A. Sloane, <a href="/A006336/a006336.pdf">Correspondence, 1977 and 1991</a>
%H A006999 James Propp and N. J. A. Sloane, <a href="/A006997/a006997.pdf">Email, March 1994</a>
%H A006999 Jane Street, <a href="https://www.janestreet.com/numberphile/">Traversing the Infinite Sidewalk</a> (2023).
%F A006999 a(n) = A061419(n) - 1.
%F A006999 a(n) = A061418(n) - 2.
%F A006999 a(n) = floor((3a(n-1)+2)/2).
%F A006999 a(n) = -1 + floor(c*(3/2)^n) where c=1.0815136... - _Benoit Cloitre_, Jan 10 2002; this constant c is 2/3*K(3) (see A083286). - _Ralf Stephan_, May 29 2003
%F A006999 a(n+1) = (3*a(n))/2+1 if a(n) is even. a(n+1) = (3*a(n)+1)/2 if a(n) is odd. - _Miquel Cerda_, Jun 15 2019
%t A006999 a[0] = 0; a[n_] := a[n] = Floor[(3 a[n-1] + 2)/2];
%t A006999 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 01 2018 *)
%o A006999 (PARI) a(n)=if(n<1,0,floor((3*a(n-1)+2)/2))
%o A006999 (Haskell)
%o A006999 a006999 n = a006999_list !! n
%o A006999 a006999_list = 0 : map ((`div` 2) . (+ 2) . (* 3)) a006999_list
%o A006999 -- _Reinhard Zumkeller_, Oct 26 2011
%Y A006999 Cf. A061419, A061418, A005428 (first differences), A083286.
%Y A006999 Cf. A006997, A016789, A152009.
%Y A006999 Cf. A003312.
%K A006999 nonn,easy
%O A006999 0,3
%A A006999 _N. J. A. Sloane_, D. R. Hofstadter, and _James Propp_, Jul 15 1977
%E A006999 More terms from _James Sellers_, Feb 06 2000