A161639 - OEIS

%I #22 Jul 23 2021 17:41:15

%S 8,9,10,11,12,13,14,15,40,41,42,43,44,45,46,47,56,57,58,59,60,61,62,

%T 63,72,73,74,75,76,77,78,79,104,105,106,107,108,109,110,111,136,137,

%U 138,139,140,141,142,143,168,169,170,171,172,173,174,175,184,185,186,187,188,189

%N Positions n such that A010060(n) = A010060(n+8).

%C Locates correlations of the form 1xxxxxxx1 or 0xxxxxxx0 in the Thue-Morse sequence.

%C Or: union of numbers 8*A079523(n)+k, k=0, 1, 2, 3, 4, 5, 6, or 7.

%C Generalization: the numbers n such that A010060(n) = A010060(n+2^m) constitute the union of sequences {2^m*A079523(n)+k}, k=0,1,...,2^m-1.

%H G. C. Greubel, <a href="/A161639/b161639.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: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="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. [_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+8], Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Oct 24 2013 *)

%t SequencePosition[ThueMorse[Range[0,200]],{x_,_,_,_,_,_,_,_,x_}][[All,1]]-1 (* _Harvey P. Dale_, Jul 23 2021 *)

%o (PARI) is(n)=hammingweight(n)%2==hammingweight(n+8)%2 \\ _Charles R Greathouse IV_, Aug 20 2013

%Y Cf. A161579, A161580, A161627, A131323, A036554, A010060, A121539, A079523, A081706

%K nonn,base

%O 1,1

%A _Vladimir Shevelev_, Jun 15 2009

%E Duplicate of 174 removed by _R. J. Mathar_, Aug 28 2009