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Partitioning integers to avoid arithmetic progressions of length 3.
(Formerly M0990)
1

%I M0990 #24 Jun 10 2021 16:02:41

%S 0,1,2,4,6,8,12,14,16,24,26,28,32,40,48,52,54,56,64,72,80,96,100,104,

%T 108,110,112,128,136,144,160,176,192,200,204,208,216,218,220,224,240,

%U 256,272,280,288,320,336,352,384,392,400,408,412,416,432,434,436,440

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

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

%H Rémy Sigrist, <a href="/A006998/b006998.txt">Table of n, a(n) for n = 0..10000</a>

%H Joseph Gerver, James Propp and Jamie Simpson, <a href="https://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., Vol. 102, No. 3 (1988), 765-772.

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

%F a(n) = a([ 2n/3 ]) + a([ (2n+1)/3 ]).

%o (PARI) for (n=1, #a=vector(58), print1 (a[n]=if (n<=2, n-1, a[1+((2*n-2)\3)]+a[1+((2*n-1)\3)])", ")) \\ _Rémy Sigrist_, Jun 10 2021

%Y Cf. A006997, A006999.

%K nonn,look

%O 0,3

%A _N. J. A. Sloane_, _James Propp_

%E More terms from _Rémy Sigrist_, Jun 10 2021