A161639 Positions n such that A010060(n) = A010060(n+8).

8, 9, 10, 11, 12, 13, 14, 15, 40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 72, 73, 74, 75, 76, 77, 78, 79, 104, 105, 106, 107, 108, 109, 110, 111, 136, 137, 138, 139, 140, 141, 142, 143, 168, 169, 170, 171, 172, 173, 174, 175, 184, 185, 186, 187, 188, 189

Offset: 1

Author

Vladimir Shevelev, Jun 15 2009

Keywords

• nonn
• base

Comments

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

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

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv:1401.3727 [math.NT], 2014.
• J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388.
• V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence arXiv:0907.0880 [math.NT], 2009-2012. [_Vladimir Shevelev_, Jul 31 2009]

Crossrefs

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

Programs

Mathematica

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 *)
SequencePosition[ThueMorse[Range[0,200]],{x_,,,_,,,_,,x}][[All,1]]-1 (* Harvey P. Dale, Jul 23 2021 *)

PARI

is(n)=hammingweight(n)%2==hammingweight(n+8)%2 \\ Charles R Greathouse IV, Aug 20 2013

Extensions

Duplicate of 174 removed by R. J. Mathar, Aug 28 2009