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A003278
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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.
(Formerly M0975)
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21
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1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, 41, 82, 83, 85, 86, 91, 92, 94, 95, 109, 110, 112, 113, 118, 119, 121, 122, 244, 245, 247, 248, 253, 254, 256, 257, 271, 272, 274, 275, 280, 281, 283, 284, 325, 326, 328, 329, 334, 335, 337, 338, 352, 353
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| That is, there are no three elements A, B and C such that B - A = C - B.
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
Sums of distinct powers of 3, translated by 1.
Positions of 0 in A189820; complement of A189822. [From Clark Kimberling, May 26 2011]
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REFERENCES
| J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
P. Erdos and P. Turan, On some sequences of integers, J. London Math. Soc., 11 (1936), 261-264.
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.
R. K. Guy, Unsolved Problems in Number Theory, E10.
Iacobescu, F. 'Smarandache Partition Type and Other Sequences.' Bull. Pure Appl. Sci. 16E, 237-240, 1997.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1024
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
M. L. Perez et al., eds., Smarandache Notions Journal
F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.
Eric Weisstein's World of Mathematics, Smarandache Sequences
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FORMULA
| a(2k + 1) = a(2k) + 1, a(2^k + 1) = 2*a(2^k).
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
G.f. 1/(1-x)^2 + sum(3^(k-1)*x^(2^k)/((1-x^(2^k))*(1-x)),k=1..infinity). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Sep 10 2003, corrected by Robert Israel, May 25 2011
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MATHEMATICA
| (* 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)
(* Next program: A003278 from A189820, from Clark Kimberling, May 26 2011 *)
a[1] = 0; h = 180;
Table[a[3 k - 2] = a[k], {k, 1, h}];
Table[a[3 k - 1] = a[k], {k, 1, h}];
Table[a[3 k] = 1, {k, 1, h}];
Table[a[n], {n, 1, h}] (*A189820*)
Flatten[Position[%, 0]] (*A003278*)
Flatten[Position[%%, 1]] (*A189822*)
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PROG
| (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
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CROSSREFS
| Equals 1 + A005836. Cf. A001511, A098871.
Row 0 of array in A093682.
Sequence in context: A173817 A198383 A156799 * A004792 A167795 A138048
Adjacent sequences: A003275 A003276 A003277 * A003279 A003280 A003281
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. P. Stanley
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