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%I #77 Mar 13 2023 09:53:48
%S 1,2,4,5,8,9,10,13,16,17,18,20,21,25,26,29,32,33,34,36,37,40,41,42,45,
%T 49,50,52,53,57,58,61,64,65,66,68,69,72,73,74,77,80,81,82,84,85,89,90,
%U 93,97,98,100,101,104,105,106,109,113,114,116,117,121,122,125,128,129
%N Numbers whose odd part is of the form 4k+1. The bit to the left of the least significant bit of each term is unset.
%C Either of form 2a(m) or 4k+1, k >= 0, 0 < m < n.
%C A000265(a(n)) is an element of A016813.
%C a(n) such that A038189(a(n)) = 0.
%C Numbers n such that kronecker(n, m) = kronecker(m, n) for all m. - _Michael Somos_, Sep 24 2005
%C The Dragon curve A014577 (but changing the offset to 1: (1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...) = the characteristic function of A091072. - _Gary W. Adamson_, Apr 11 2010
%C A014707(a(n) + 1) = 0. - _Reinhard Zumkeller_, Sep 28 2011
%C A055975(a(n)) > 0. - _Reinhard Zumkeller_, Apr 28 2012
%C Also indices of 1 in A034947. - _Jianing Song_, Apr 24 2021
%C The terms in the sequence are the same as the terms in the odd columns of the table in A135764 with headings 4k+1: (1, 5, 9, 13...). A014577(n) = 1 if n is in that set, but A014577(n) = 0 if n is in the set of even columns in the A135764 table. - _Gary W. Adamson_, May 29 2021
%H Reinhard Zumkeller, <a href="/A091072/b091072.txt">Table of n, a(n) for n = 1..10000</a>
%H J.-P. Allouche and J. Shallit, <a href="https://arxiv.org/abs/2006.04708">On three conjectures of P. Barry</a>, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
%H Paul Barry, <a href="https://arxiv.org/abs/2005.04066">Some observations on the Rueppel sequence and associated Hankel determinants</a>, arXiv:2005.04066 [math.CO], 2020.
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index TurnLeft, with a(n) = TurnLeft(n-1).
%H J. E. S. Socolar and J. M. Taylor, <a href="http://arxiv.org/abs/1003.4279">An aperiodic hexagonal tile</a>, arXiv:1003.4279 [math.CO], 2010.
%e x + 2*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 13*x^8 + 16*x^9 + ...
%t Select[ Range[129], EvenQ[ (#/2^IntegerExponent[#, 2] - 1)/2 ] & ] (* _Jean-François Alcover_, Feb 16 2012, after Pari *)
%o (PARI) for(n=1,200,if(((n/2^valuation(n,2)-1)/2)%2==0,print1(n",")))
%o (PARI) {a(n) = local(m, c); if( n<1, 0, c=1; m=1; while( c<n, m++; if( ((m / 2^valuation( m,2) - 1) / 2)%2==0, c++)); m)} /* _Michael Somos_, Sep 24 2005 */
%o (PARI) a(n) = if(n=2*n-2, my(t=1); forstep(i=logint(n,2),0,-1, if(bittest(n,i)==t, n--;t=!t))); n+1; \\ _Kevin Ryde_, Mar 21 2021
%o (Haskell)
%o import Data.List (elemIndices)
%o a091072 n = a091072_list !! (n-1)
%o a091072_list = map (+ 1) $ elemIndices 0 a014707_list
%o -- _Reinhard Zumkeller_, Sep 28 2011
%Y Complement of A091067.
%Y Cf. A014577 (characteristic function), A034947, A106841 (first of triplet), A088742 (first differences), A339597.
%K nonn,easy
%O 1,2
%A _Ralf Stephan_, Feb 22 2004
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