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A161627 Positions n such that A010060(n)=A010060(n+4). 13

%I #20 Apr 16 2018 02:52:24

%S 4,5,6,7,20,21,22,23,28,29,30,31,36,37,38,39,52,53,54,55,68,69,70,71,

%T 84,85,86,87,92,93,94,95,100,101,102,103,116,117,118,119,124,125,126,

%U 127,132,133,134,135,148,149,150,151,156,157,158,159,164,165,166,167,180,181,182

%N Positions n such that A010060(n)=A010060(n+4).

%C Or: union of the numbers of the form 4*A079523(n)+k, k=0, 1, 2, or 3.

%C Locates patterns of the form 1xxx1 or 0xxx0 in the Thue-Morse sequence.

%H Harvey P. Dale, <a href="/A161627/b161627.txt">Table of n, a(n) for n = 1..1000</a>

%H J.-P. Allouche, <a href="https://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 V. Shevelev, <a href="https://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. [From _Vladimir Shevelev_, Jul 31 2009]

%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 <= 200, n++, If[tm[n] == tm[n+4], Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Oct 24 2013 *)

%t SequencePosition[ThueMorse[Range[200]],{x_,_,_,_,x_}][[All,1]] (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Apr 16 2017 *)

%o (PARI) is(n)=hammingweight(n)%2==hammingweight(n+4)%2 \\ _Charles R Greathouse IV_, Aug 20 2013

%Y Cf. A161579, A161580, A079523, A131323, A036554, A010060, A121539, A079523, A081706.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Jun 15 2009

%E Extended by _R. J. Mathar_, Aug 28 2009

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