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%I #21 Jan 30 2021 04:23:15
%S 0,1,4,5,6,7,8,9,12,13,16,17,20,21,22,23,24,25,28,29,30,31,32,33,36,
%T 37,38,39,40,41,44,45,48,49,52,53,54,55,56,57,60,61,64,65,68,69,70,71,
%U 72,73,76,77,80,81,84,85,86,87,88,89,92,93,94,95,96,97,100,101,102,103,104
%N Positions n such that A010060(n) + A010060(n+2) = 1.
%C Locates patterns of the form 0x1 or 1x0 in the Thue-Morse sequence.
%C Complement to A081706. Also: union of sequences {2*A121539(n)+k}, k=0 or 1, generalized in A161673.
%C Also union of sequences {A079523(n)-k}, k=0 or 1. For a generalization see A161890. - _Vladimir Shevelev_, Jul 05 2009
%C The asymptotic density of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - _Amiram Eldar_, Jan 30 2021
%H G. C. Greubel, <a href="/A161674/b161674.txt">Table of n, a(n) for n = 1..10000</a>
%H Rob Burns, <a href="https://arxiv.org/abs/1611.04910">Asymptotic density of Motzkin numbers modulo small primes</a>, arXiv:1611.04910 [math.NT], 2016.
%H Eric Rowland and Reem Yassawi, <a href="http://www.numdam.org/item/JTNB_2015__27_1_245_0/">Automatic congruences for diagonals of rational functions</a>, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.
%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 <= 6000, n++, If[tm[n] + tm[n + 2] == 1, Sow[n]]]][[2, 1]] (* _G. C. Greubel_, Jan 05 2018 *)
%t Flatten[Position[Partition[ThueMorse[Range[0,120]],3,1],_?(#[[1]]+#[[3]] == 1&),1,Heads->False]]-1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 29 2019 *)
%o (PARI) is(n)=hammingweight(n)%2!=hammingweight(n+2)%2 \\ _Charles R Greathouse IV_, Aug 20 2013
%Y Cf. A161673, A161639, A161641, A161627, A161579, A161580, A161890, A121539, A131323, A036554, A010060, A079523, A081706.
%K nonn,easy
%O 1,3
%A _Vladimir Shevelev_, Jun 16 2009
%E Extended by _R. J. Mathar_, Aug 28 2009
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