A161579 -id:A161579 - cp's OEIS frontend

A161974 a(n) = number of equalities of the form A010060(n+k) = A010060(n), k=1,2,3.

1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1

Offset: 0

Vladimir Shevelev, Jun 23 2009

See comment to A161916. 3-a(n) is the number of equalities of kind A010060(n+k) = 1-A010060(n), k=1,2,3.

Cf. A161916, A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060.

a(n)=sum(k=1,3,hammingweight(n)%2==hammingweight(n+k)%2) \\ Charles R Greathouse IV, Aug 20 2013

Missing a(24)=1 inserted by Georg Fischer, Jun 21 2024

A162311 Numbers such that A010060(n) = A010060(n+7).

Original entry on oeis.org

1, 3, 4, 5, 7, 10, 14, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 42, 46, 49, 51, 52, 53, 55, 58, 62

Offset: 1

Vladimir Shevelev, Jul 01 2009

Or union of intersection of A161673 and {A121539(n)-7} and intersection of A161639 and {A079523(n)-7}.

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

Charles R Greathouse IV, 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.

Cf. A161890, A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706, A161916, A161974.

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 <= 20000, n++, If[tm[n] == tm[n + 7], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n)%2==hammingweight(n+7)%2 \\ Charles R Greathouse IV, Aug 20 2013

A162648 Locations of patterns 1001 or 0110 in the Thue-Morse sequence A010060.

Original entry on oeis.org

0, 4, 6, 8, 12, 16, 20, 22, 24, 28, 30, 32, 36, 38, 40, 44, 48, 52, 54, 56, 60, 64, 68, 70, 72, 76, 80, 84, 86, 88, 92, 94, 96, 100, 102, 104, 108, 112, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 144, 148, 150, 152, 156, 158, 160, 164, 166, 168, 172, 176, 180

Offset: 1

Vladimir Shevelev, Jul 08 2009

nonn

Numbers n for which A010060(n+1) = A010060(n+2) = 1-A010060(n) and A010060(n+3) = A010060(n).

Or intersection of A121539, A161674, and A161579.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

Select[Range[3500], MatchQ[IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[Length[{b}]]] &] - 1 (* G. C. Greubel, Jan 05 2018 *)
With[{nn=200},Sort[Join[SequencePosition[ThueMorse[Range[0,nn]],{1,0,0,1}],SequencePosition[ ThueMorse[Range[0,nn]],{0,1,1,0}]]][[;;,1]]]-1 (* Harvey P. Dale, Aug 20 2024 *)

is(n)=my(v=vector(4,i,hammingweight(n+i-1))); v[1]==v[4] && v[1]!=v[2] && v[1]!=v[3] \\ Charles R Greathouse IV, Aug 20 2013

a(n) = A079523(n) - 1.

More readable definition from R. J. Mathar, Sep 16 2009

cp's OEIS Frontend

A161974 a(n) = number of equalities of the form A010060(n+k) = A010060(n), k=1,2,3.

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A162311 Numbers such that A010060(n) = A010060(n+7).

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A162648 Locations of patterns 1001 or 0110 in the Thue-Morse sequence A010060.

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