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%I #27 Feb 06 2023 17:06:23
%S 0,2,3,4,6,7,9,13,15,16,18,19,20,22,24,26,27,28,30,32,34,35,36,38,39,
%T 41,45,47,48,50,51,52,54,55,57,61,63,64,66,67,68,70,71,73,77,79,80,82,
%U 83,84,86,88,90,91,92,94,96,98,99,100,102,103,105,109,111,112,114,115,116,118,120
%N Numbers such that A010060(n) = A010060(n+9)
%C Or union of intersection of A161639 and {A079523(n)-8} and intersection of A161673 and {A121539(n)-8}. In general, for a>=1, consider equations A010060(x+a)+A010060(x)=1, A010060(x+a)=A010060(x). Denote via B_a (C_a) the sequence of nonnegative solutions of the first (second) equation. Then we have recursions: B_(a+1) is the union of transactions 1) C_a and {A121539(n)-a}, 2) B_a and {A079523(n)-a}; C_(a+1) is the union of transactions 1) C_a and {A079523(n)-a}, 2) B_a and {A121539(n)-a}.
%C Conjecture. In every sequence of numbers n, such that A010060(n)=A010060(n+k), for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. - _Vladimir Shevelev_, Jul 31 2009
%C This conjecture was actually proved in a later version of the Shevelev arxiv article cited below, and it can also easily be proved by the Walnut prover. - _Jeffrey Shallit_, Oct 12 2022
%H G. C. Greubel, <a href="/A161890/b161890.txt">Table of n, a(n) for n = 0..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: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 Vladimir Shevelev, <a href="http://arXiv.org/abs/0907.0880">Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence</a>, arXiv:0907.0880[math.NT], 2009-2012.
%t tm[0] = 0; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 1 - tm[(n - 1)/2]; Reap[For[n = 0, n <= 18000, n++, If[tm[n] == tm[n + 9], Sow[n]]]][[2, 1]] (* _G. C. Greubel_, Jan 05 2018 *)
%t SequencePosition[ThueMorse[Range[0,150]],{x_,_,_,_,_,_,_,_,_,x_}][[All,1]]-1 (* _Harvey P. Dale_, Feb 06 2023 *)
%o (PARI) is(n)=hammingweight(n)%2==hammingweight(n+9)%2 \\ _Charles R Greathouse IV_, Aug 20 2013
%Y Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.
%K nonn,base
%O 0,2
%A _Vladimir Shevelev_, Jun 21 2009
%E Terms a(35) onward added by _G. C. Greubel_, Jan 05 2018
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