Search: id:A003278
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%I A003278 M0975
%S A003278 1,2,4,5,10,11,13,14,28,29,31,32,37,38,40,41,82,83,85,86,91,92,94,95,
%T A003278 109,110,112,113,118,119,121,122,244,245,247,248,253,254,256,257,271,
%U A003278 272,274,275,280,281,283,284,325,326,328,329,334,335,337,338,352,353
%N A003278 a(n)-1 in ternary = n-1 in binary; also: a(1) = 1, a(2) = 2, a(n) is 
               smallest number k which avoids any 3-term arithmetic progression 
               in a(1), a(2), ..., a(n-1), k.
%C A003278 That is, there are no three elements A, B and C such that B - A = C - 
               B.
%C A003278 Difference sequence related to Gray code bit sequence (A001511). The 
               difference patterns follows a similar repeating pattern (ABACABADABACABAE...), 
               but each new value is the sum of the previous values, rather than 
               simply 1 more than the maximum of the previous values. - Hal Burch 
               (hburch(AT)cs.cmu.edu), Jan 12 2004
%C A003278 Sums of distinct powers of 3, translated by 1.
%D A003278 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical 
               Computer Sci., 98 (1992), 163-197.
%D A003278 P. Erdos and P. Turan, On some sequences of integers, J. London Math. 
               Soc., 11 (1936), 261-264.
%D A003278 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.
%D A003278 R. K. Guy, Unsolved Problems in Number Theory, E10.
%D A003278 Iacobescu, F. 'Smarandache Partition Type and Other Sequences.' Bull. 
               Pure Appl. Sci. 16E, 237-240, 1997.
%D A003278 H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, 
               Vol. 8, No. 1-2-3, 1997, 170-183.
%D A003278 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003278 F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, 
               Phoenix, 2006.
%H A003278 T. D. Noe, <a href="b003278.txt">Table of n, a(n) for n=1..1024</a>
%H A003278 J.-P. Allouche and J. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/
               Papers/as0.ps">The ring of k-regular sequences</a>, Theoretical Computer 
               Sci., 98 (1992), 163-197.
%H A003278 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
               ">Smarandache Notions Journal</a>
%H A003278 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/Sequences-book.pdf">
               Sequences of Numbers Involved in Unsolved Problems</a>.
%H A003278 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SmarandacheSequences.html">Link to a section of The World of Mathematics.</
               a>
%F A003278 a(2k + 1) = a(2k) + 1, a(2^k + 1) = 2*a(2^k).
%F A003278 a(n) = b(n+1) with b(0)=1, b(2n)=3b(n)-2, b(2n+1)=3b(n)-1. - Ralf Stephan 
               (ralf(AT)ark.in-berlin.de), Aug 23 2003
%F A003278 G.f. 1/(1-x) * (1 + sum(k>=0, 3^k/(1+x^2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Sep 10 2003
%t A003278 (* first do *) Needs["DiscreteMath`Combinatorica`"]; (* then *) Take[ 
               Sort[ Plus @@@ Subsets[ Table[3^n, {n, 0, 6}]]] + 1, 58] (from Robert 
               G. Wilson v Oct 23 2004)
%o A003278 #!/usr/bin/perl $nxt = 1; @list = (); for ($cnt = 0; $cnt < 1500; $cnt++) 
               { while (exists $legal{$nxt}) { $nxt++; } print "$nxt "; last if 
               ($nxt >= 1000000); for ($i = 0; $i <= $#list; $i++) { $t = 2*$nxt 
               - $list[$i]; $legal{$t} = -1; } $cnt++; push @list, $nxt; $nxt++; 
               } (Hal Burch)
%Y A003278 Equals 1 + A005836. Cf. A001511, A098871.
%Y A003278 Row 0 of array in A093682.
%Y A003278 Sequence in context: A122991 A125728 A156799 this_sequence A004792 A167795 
               A138048
%Y A003278 Adjacent sequences: A003275 A003276 A003277 this_sequence A003279 A003280 
               A003281
%K A003278 nonn,nice,easy
%O A003278 1,2
%A A003278 N. J. A. Sloane (njas(AT)research.att.com), R. P. Stanley

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