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A003159
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Numbers n such that binary representation ends in even number of zeros.
(Formerly M2306)
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41
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1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Fraenkel (2010) called these the "vile" numbers.
Minimal with respect to property that parity of number of 1's in binary expansion alternates.
Minimal with respect to property that sequence is half its complement. [Corrected by Aviezri Fraenkel, Jan 29 2010]
If n appears then 2n does not.
Increasing sequence of positive integers n such that A035263(n)=1 (from paper by Allouche et al.). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2003
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 16 2004
Indices of odd numbers in A007913, in A001511. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 27 2004
Partial sums of A026465. - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
A121701(2*a(n)) = A121701(a(n)); A096268(a(n)-1) = 0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 16 2006
A different permutation of the same terms may be found in A141290 and the accompanying array. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2008
a(n) = n-th clockwise Tower of Hanoi move; counterclockwise if not in the sequence. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 14 2008
Indices of terms of Thue-Morse sequence A010060 which are different from the previous term. [From Tanya Khovanova (tanyakh(AT)yahoo.com), Jan 06 2009]
The sequence has the following fractal property. Remove odd terms from the sequence, it remains even terms 4,12,16,20,28,36,44,48,52,... Divides these terms by 4 we get 1,3,4,5,7,9,11,12,...the original sequence. [From Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 06 2010]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2010: (Start)
A conjectured identity relating to the partition sequence, A000041 as polcoeff
p(x); A003159, and its characteristic function A035263: (1, 0, 1, 1, 1, 0,
1, 0, 1, 0, 1, 1, 1,...); and A036554 indicating n-th terms with zeros in A035263: (2, 6, 8, 10, 14, 18, 22,...).
The conjecture states that p(x) = A(x) = A(x^2) when A(x) = polcoeff A174065
= the Euler transform of A035263 = 1/(1-x)*(1-x^3)*(1-x^4)*(1-x^5)* ...=
(1 + x + x^2 + 2x^3 + 3x^4 + 4x^5 + ...) and the aerated variant = the Euler
transform of the complement of A035263: 1/(1-x^2)*(1-x^6)*(1-x^8)* ... =
(1 + x^2 + x^4 + 2x^6 + 3x^8 + 4x^10 + ...). (End)
If the lower s-Wythoff sequence of s is s, then s=A003159. (See A184117 for the definition of lower and upper s-Wythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A003159 is the limit when the lower s-Wythoff operation is iterated. For example, starting with s=(1,4,9,16,...)=(n^2), we obtain lower and upper s-Wythoff sequences
a=(1,3,4,5,6,8,9,10,11,12,14...)=A184427;
b=(2,7,12,21,31,44,58,74,...)=A184428.
Then putting s=a and repeating the operation gives
a'=(1,3,4,5,7,9,11,12,14,...), which has the same first eight terms as A003159. [From Clark Kimberling, ck6(AT)evansville.edu Jan 14 2011]
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REFERENCES
| L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Representations for a special sequence, Fib. Quart., 10 (1972), 499-518, 550.
Michael Domaratzki, Trajectory-based codes, Acta Informatica, Volume 40, Numbers 6-7 / May, 2004. [From N. J. A. Sloane, Jul 10 2009]
Aviezri S. Fraenkel, The vile, dopy, evil and odious game players, Discrete Math., to appear (2010).
C. Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160. [From N. J. A. Sloane, Jan 31 2012]
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Problem E2850, Amer. Math. Monthly, 87 (1980), 671.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.
J.-P. Allouche and J. O. Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
A. S. Fraenkel, Home Page
Index entries for sequences related to binary expansion of n
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FORMULA
| a(0)=1; for n>=0, a(n+1) = a(n)+1 if (a(n)+1)/2 is not already in the sequence, = a(n)+2 otherwise.
Lim n ->infinity a(n)/n = 3/2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 13 2002
a(n)=sum(k=1, n, A026465(k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 31 2003
a(n+1) = (if a(n) mod 4 = 3 then A007814(a(n) + 1) mod 2 else a(n) mod 2) + a(n) + 1; a(1) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 03 2003
a(A003157(n)) is even. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 22 2004
Sequence consists of numbers of the form 4^i*(2j+1), i>=0, j>=0. - Jon Perry (perry(AT)globalnet.co.uk), Jun 06 2004
G.f.: (1/(1-x))Product{k>=1, 1+x^A001045(k)}. - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
a(1)=1, a(2)=3 and for n>=2 we get a(n+1)=4+a(n)+a(n-1)-a(a(n)-n+1)-a(a(n-1)-n+2) [From Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 08 2010]
If A(x) is the counting function for a(n)<=x, then A(2^n)=(2^(n+1)+(-1)^n)/3 [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 15 2010]
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EXAMPLE
| 1=1 (odd), 3=11 (even), 4=100 (odd), 5=101 (even), 7=111 (odd), ...
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MATHEMATICA
| f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; c]; Select[ Range[105], EvenQ[ f[ # ]] & ]
Select[Range[150], EvenQ[IntegerExponent[#, 2]]&] (* From Harvey P. Dale, Oct 19 2011 *)
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PROG
| (PARI) a(n)=if(n<2, n>0, n=a(n-1); until(valuation(n, 2)%2==0, n++); n)
(Haskell)
import Data.List (delete)
a003159 n = a003159_list !! (n-1)
a003159_list = f [1..] where f (x:xs) = x : f (delete (2*x) xs)
-- Reinhard Zumkeller, Nov 04 2011
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CROSSREFS
| Indices of even numbers in A007814. Complement of A036554, also one-half of A036554. Cf. A001285, A010060.
Equals A056196(n)/8.
Cf. A000041, A174065 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2010]
Sequence in context: A156246 A136014 A112930 * A187691 A141259 A047501
Adjacent sequences: A003156 A003157 A003158 * A003160 A003161 A003162
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KEYWORD
| nonn,nice,easy,eigen,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Additional comments from Michael Somos.
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