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 A006999 Partitioning integers to avoid arithmetic progressions of length 3. (Formerly M1047) 17
 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; text; internal format)
 OFFSET 0,3 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. - John W. Layman, Nov 20 2008 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 B. Chen, R. Chen, J. Guo, S. Lee et al, On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018. Joseph Gerver, James Propp and Jamie Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772. D. R. Hofstadter, Eta-Lore [Cached copy, with permission] D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission] D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991 James Propp and N. J. A. Sloane, Email, March 1994 Jane Street, Traversing the Infinite Sidewalk (2023). FORMULA a(n) = floor((3a(n-1)+2)/2). a(n) = -1 + floor(c*(3/2)^n) where c=1.081513668589844877304633988599549408710737041542024954790295591585622666484989650922411026555488940... - Benoit Cloitre, Jan 10 2002 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 MATHEMATICA a[0] = 0; a[n_] := a[n] = Floor[(3 a[n-1] + 2)/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 01 2018 *) 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 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. - John W. Layman, Nov 20 2008 Cf. A005428 (first differences). Sequence in context: A117276 A035295 A289010 * A005252 A023430 A023429 Adjacent sequences: A006996 A006997 A006998 * A007000 A007001 A007002 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, D. R. Hofstadter, and James Propp, Jul 15 1977 EXTENSIONS More terms from James A. Sellers, Feb 06 2000 STATUS approved

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Last modified August 15 12:22 EDT 2024. Contains 375173 sequences. (Running on oeis4.)