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A079523 Utterly odd numbers: numbers whose binary representation ends in an odd number of ones. 32

%I

%S 1,5,7,9,13,17,21,23,25,29,31,33,37,39,41,45,49,53,55,57,61,65,69,71,

%T 73,77,81,85,87,89,93,95,97,101,103,105,109,113,117,119,121,125,127,

%U 129,133,135,137,141,145,149,151,153,157,159,161,165,167,169,173,177,181

%N Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.

%C Also, n such that A010060(n) = A010060(n+1) where A010060 is the Thue-Morse sequence.

%C Sequence of n such that a(n) = 3n begins 7, 23, 27, 29, 31, 39, 71, 87, 91, 93, 95, ...

%C Values of k such that the Motzkin number A001006(2k) is even. Values of k such that the number of restricted hexagonal polyominoes with 2k+1 cells is even (see A002212). Values of k such that the number of directed animals of size k+1 is even (see A005773). Values of k such that the Riordan number A005043(k) is even. - _Emeric Deutsch_ and _Bruce E. Sagan_, Apr 02 2003

%C a(n) = A036554(n)-1 = A072939(n)-2. - _Ralf Stephan_, Jun 09 2003

%C Odious and evil terms alternate. - _Vladimir Shevelev_, Jun 22 2009

%C The sequence has the following fractal property: remove terms of the form 4k+1 from the sequence, and the remaining terms are of the form 4k+3: 7, 23, 31, 39, 55, 71, 87, ...; then subtract 3 from each of these terms and divide by 4 and you get the original sequence: 1, 5, 7, 9, 13, ... - _Benoit Cloitre_, Apr 06 2010

%C A035263(a(n)) = 0. - _Reinhard Zumkeller_, Mar 01 2012

%H Reinhard Zumkeller, <a href="/A079523/b079523.txt">Table of n, a(n) for n = 1..10000</a>

%H J.-P. Allouche, <a href="http://arxiv.org/abs/1401.3727">Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence</a>, arXiv preprint arXiv:1401.3727 [math.NT], 2014.

%H J.-P. Allouche, <a href="http://dx.doi.org/10.5802/jtnb.906">Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence</a>, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.

%H J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, Simon Plouffe, B. E. Sagan, <a href="http://dx.doi.org/10.1016/0012-365X(93)00147-W">A relative of the Thue-Morse sequence</a>, Discrete Math., 139, 1995, 455-461.

%H Narad Rampersad, Manon Stipulanti, <a href="https://arxiv.org/abs/1807.11899">The Formal Inverse of the Period-Doubling Sequence</a>, arXiv:1807.11899 [math.CO], 2018.

%H Thomas Zaslavsky, <a href="/A075326/a075326_2.pdf">Anti-Fibonacci Numbers: A Formula</a>, Sep 26 2016 [Introduces the name "utterly odd". - _N. J. A. Sloane_, Sep 27 2016]

%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.

%F a(n) is asymptotic to 3n.

%F a(n) = 2*A003159(n) - 1. a(1)=1, a(n) = a(n-1) + 2 if (a(n-1)+1)/2 does not belong to the sequence and a(n)=a(n-1)+4 otherwise. - _Emeric Deutsch_ and Bruce E. Sagan, Apr 02 2003

%F a(n) = (1/2)A081706(2n-1).

%F a(n) = A003158(n) - n = A003157(n) - n - 1. - _Philippe Deléham_, Feb 22 2004

%F Values of k such that A091297(k) = 0. - _Philippe Deléham_, Feb 25 2004

%t Select[ Range[200], MatchQ[ IntegerDigits[#, 2], {b : (1) ..} | {___, 0, b : (1) ..} /; OddQ[ Length[{b}]]] & ] (* _Jean-François Alcover_, Jun 17 2013 *)

%o (Haskell)

%o import Data.List (elemIndices)

%o a079523 n = a079523_list !! (n-1)

%o a079523_list = elemIndices 0 a035263_list

%o -- _Reinhard Zumkeller_, Mar 01 2012

%o (PARI) is(n)=valuation(n+1,2)%2 \\ _Charles R Greathouse IV_, Mar 07 2013

%o (MAGMA) [n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 + 1]; // _Vincenzo Librandi_, Apr 16 2015

%Y Cf. A003159, A003157, A003158, A075326, A131323.

%K nonn,base,easy

%O 1,2

%A _Benoit Cloitre_, Jan 21 2003

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Last modified November 12 19:25 EST 2019. Contains 329078 sequences. (Running on oeis4.)