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A005823
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Numbers whose ternary expansion contains no 1's.
(Formerly M1567)
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52
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0, 2, 6, 8, 18, 20, 24, 26, 54, 56, 60, 62, 72, 74, 78, 80, 162, 164, 168, 170, 180, 182, 186, 188, 216, 218, 222, 224, 234, 236, 240, 242, 486, 488, 492, 494, 504, 506, 510, 512, 540, 542, 546, 548, 558, 560, 564, 566, 648, 650, 654, 656, 666, 668, 672, 674
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OFFSET
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1,2
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COMMENTS
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The set of real numbers between 0 and 1 that contain no 1's in their ternary expansion is the well-known Cantor set with Hausdorff dimension log 2 / log 3.
Complement of A081606. - Reinhard Zumkeller, Mar 23 2003
Numbers n such that the n-th Apery number equals 1 (mod 3) (cf. A005258) - Benoit Cloitre, Nov 30 2003
Numbers n such that the n-th central Delannoy number equals 1 (mod 3) (cf. A001850) - Benoit Cloitre, Nov 30 2003
Numbers n such that there exists a permutation p_1, ..., p_n of 1, ..., n such that i + p_i is a power of 3 for every i - Ray Chandler, Aug 03 2004
Subsequence of A125292; A125291(a(n)) = 1 for n>0. - Reinhard Zumkeller, Nov 26 2006
A062756(a(n)) = 0. - Reinhard Zumkeller, Mar 02 2008
The first 2^n terms of the sequence could be obtained using the Cantor process for the segment [0,3^n-1]. E.g. for n=2 we have [0,{1},2,{3,4,5},6,{7},8]. The numbers outside of braces are the first 4 terms of the sequence. Therefore the terms of the sequence could be called "Cantor's numbers". - Vladimir Shevelev, Jun 13 2008
Mahler proved that positive a(n) is never a square. - Michel Marcus, Nov 12 2012
Define t: Z -> P(R) so that t(n) is the translated Cantor ternary set spanning [n, n+1], and let T be the union of t(a(n)) for all n. T = T * 3 = T / 3 is the closure of the Cantor ternary set under multiplication by 3. - Peter Munn, Oct 30 2019
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REFERENCES
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K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 14.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
Sajed Haque, Chapter 3.4 of Discriminators of Integer Sequences, 2017, See p. 45.
Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.
C. Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160.
K. Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), 99-106.
M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, Theoret. Comput. Sci., 65 (1989), 213-220.
Eric Weisstein's World of Mathematics, Cantor Set
Index entries for 3-automatic sequences.
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FORMULA
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a(n) = 2 * A005836(n).
a(2n) = 3*a(n)+2, a(2n+1) = 3*a(n+1), a(1) = 0.
a(n) = Sum_{k = 1..n} 1 + 3^A007814(k). - Philippe Deléham, Jul 09 2005
If the offset were changed to zero, then: a(0)=0, a(n+1)=f(a(n)+1,f(a(n)+1) where f(x,y) = if x<3 and x<>1 then y else if x mod 3 = 1 then f(y+1,y+1) else f(floor(x/3),y). - Reinhard Zumkeller, Mar 02 2008
G.f. g(x) satisfies g(x) = 3*g(x^2)*(1+1/x) + 2*x^2/(1-x^2). - Robert Israel, Jan 04 2015
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MAPLE
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a:= proc(n) option remember;
`if`(n=1, 0, `if`(irem (n, 2, 'q')=0, 3*a(q)+2, 3*a(q+1)))
end:
seq(a(n), n=1..100); # Alois P. Heinz, Apr 19 2012
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MATHEMATICA
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Select[ Range[ 0, 729 ], (Count[ IntegerDigits[ #, 3 ], 1 ]==0)& ]
Select[Range[0, 700], DigitCount[#, 3, 1]==0&] (* Harvey P. Dale, Mar 12 2016 *)
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PROG
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(PARI) is(n)=while(n, if(n%3==1, return(0), n\=3)); 1 \\ Charles R Greathouse IV, Apr 20 2012
(PARI) a(n)=n=binary(n-1); sum(i=1, #n, 2*n[i]*3^(#n-i)) \\ Charles R Greathouse IV, Apr 20 2012
(PARI) a(n)=2*fromdigits(binary(n-1), 3) \\ Charles R Greathouse IV, Aug 24 2016
(Python)
def A005823(n):
return 2*int(format(n-1, 'b'), 3) # Chai Wah Wu, Jan 04 2015
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CROSSREFS
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Twice A005836.
Cf. A032924, A014263, A007089, A062756, A061392, A001196, A097252-A097262.
Cf. A088917 (characteristic function), A306556.
Sequence in context: A325686 A053355 A233572 * A259026 A178758 A024431
Adjacent sequences: A005820 A005821 A005822 * A005824 A005825 A005826
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KEYWORD
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nonn,easy,nice,look,base
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AUTHOR
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N. J. A. Sloane, Jeffrey Shallit
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EXTENSIONS
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More terms from Sascha Kurz, Mar 24 2002
Offset corrected by N. J. A. Sloane, Mar 02 2008. This may require some of the formulas to be adjusted.
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STATUS
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approved
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