A006999 - OEIS

%I M1047 #63 Jul 30 2023 18:46:38

%S 0,1,2,4,7,11,17,26,40,61,92,139,209,314,472,709,1064,1597,2396,3595,

%T 5393,8090,12136,18205,27308,40963,61445,92168,138253,207380,311071,

%U 466607,699911,1049867,1574801,2362202,3543304,5314957,7972436

%N Partitioning integers to avoid arithmetic progressions of length 3.

%C a(n) = A006997(3^n-1).

%C 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 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A006999/b006999.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 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 D. R. Hofstadter, <a href="/A006336/a006336_1.pdf">Eta-Lore</a> [Cached copy, with permission]

%H D. R. Hofstadter, <a href="/A006336/a006336_2.pdf">Pi-Mu Sequences</a> [Cached copy, with permission]

%H D. R. Hofstadter and N. J. A. Sloane, <a href="/A006336/a006336.pdf">Correspondence, 1977 and 1991</a>

%H James Propp and N. J. A. Sloane, <a href="/A006997/a006997.pdf">Email, March 1994</a>

%H Jane Street, <a href="https://www.janestreet.com/numberphile/">Traversing the Infinite Sidewalk</a> (2023).

%F a(n) = floor((3a(n-1)+2)/2).

%F a(n) = -1 + floor(c*(3/2)^n) where c=1.081513668589844877304633988599549408710737041542024954790295591585622666484989650922411026555488940... - _Benoit Cloitre_, Jan 10 2002

%F 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 a[0] = 0; a[n_] := a[n] = Floor[(3 a[n-1] + 2)/2];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 01 2018 *)

%o (PARI) a(n)=if(n<1,0,floor((3*a(n-1)+2)/2))

%o (Haskell)

%o a006999 n = a006999_list !! n

%o a006999_list = 0 : map ((`div` 2) . (+ 2) . (* 3)) a006999_list

%o -- _Reinhard Zumkeller_, Oct 26 2011

%Y a(n) = A061419(n) - 1 = A061418(n) - 2.

%Y The constant c is 2/3*K(3) (see A083286). - _Ralf Stephan_, May 29 2003

%Y Cf. A003312.

%Y First differences are in A073941.

%Y Cf. A016789, A152009. - _John W. Layman_, Nov 20 2008

%Y Cf. A005428 (first differences).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, D. R. Hofstadter, and _James Propp_, Jul 15 1977

%E More terms from _James A. Sellers_, Feb 06 2000