A161579 -id:A161579 - cp's OEIS frontend

A161580 Positions n such that A010060(n) + A010060(n+3) = 1.

2, 5, 7, 10, 14, 18, 21, 23, 26, 29, 31, 34, 37, 39, 42, 46, 50, 53, 55, 58, 62, 66, 69, 71, 74, 78, 82, 85, 87, 90, 93, 95, 98, 101, 103, 106, 110, 114, 117, 119, 122, 125, 127, 130, 133, 135, 138, 142, 146, 149, 151, 154, 157, 159, 162, 165, 167, 170, 174, 178, 181, 183, 186

Offset: 1

Vladimir Shevelev, Jun 14 2009

nonn

Conjecture: In every sequence of numbers n such that A010060(n) + A010060(n+k) = 1, for fixed odd k, the odious (A000069) and evil (A001969) terms alternate. [From Vladimir Shevelev, Jul 31 2009]

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

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

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+3] == 1, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 24 2013 *)

is(n)=hammingweight(n)%2+hammingweight(n+3)%2==1 \\ Charles R Greathouse IV, Mar 22 2013

A001477 \ A161579.

More terms from R. J. Mathar, Aug 17 2009

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

Original entry on oeis.org

4, 5, 6, 7, 20, 21, 22, 23, 28, 29, 30, 31, 36, 37, 38, 39, 52, 53, 54, 55, 68, 69, 70, 71, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 116, 117, 118, 119, 124, 125, 126, 127, 132, 133, 134, 135, 148, 149, 150, 151, 156, 157, 158, 159, 164, 165, 166, 167, 180, 181, 182

Offset: 1

Vladimir Shevelev, Jun 15 2009

nonn

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

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

Harvey P. Dale, Table of n, a(n) for n = 1..1000
J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, arXiv preprint 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. [From _Vladimir Shevelev_, Jul 31 2009]

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

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

is(n)=hammingweight(n)%2==hammingweight(n+4)%2 \\ Charles R Greathouse IV, Aug 20 2013

Extended by R. J. Mathar, Aug 28 2009

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

Original entry on oeis.org

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

Vladimir Shevelev, Jun 15 2009

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.

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]

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

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 *)

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

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

A161641 Positions n such that A010060(n) + A010060(n+4) = 1.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99, 104, 105, 106, 107, 108

Offset: 1

Vladimir Shevelev, Jun 15 2009

nonn

Also union of all numbers of the form A131323(n)-k, k=0, 1, 2, or 3.

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

Cf. A161639, A161627, A161579, A161580, A079523, A131323, A036554, A010060, A121539, A079523, A081706.

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 <= 16000, n++, If[tm[n] + tm[n + 4] == 1, Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 01 2018 *)

is(n)=hammingweight(n)%2!=hammingweight(n+4)%2 \\ Charles R Greathouse IV, Aug 20 2013

A001477 \ A161627.

More terms from R. J. Mathar, Aug 17 2009

A161674 Positions n such that A010060(n) + A010060(n+2) = 1.

Original entry on oeis.org

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, 37, 38, 39, 40, 41, 44, 45, 48, 49, 52, 53, 54, 55, 56, 57, 60, 61, 64, 65, 68, 69, 70, 71, 72, 73, 76, 77, 80, 81, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 97, 100, 101, 102, 103, 104

Offset: 1

Vladimir Shevelev, Jun 16 2009

Locates patterns of the form 0x1 or 1x0 in the Thue-Morse sequence.
Complement to A081706. Also: union of sequences {2*A121539(n)+k}, k=0 or 1, generalized in A161673.
Also union of sequences {A079523(n)-k}, k=0 or 1. For a generalization see A161890. - Vladimir Shevelev, Jul 05 2009
The asymptotic density of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021

G. C. Greubel, Table of n, a(n) for n = 1..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.

Cf. A161673, A161639, A161641, A161627, A161579, A161580, A161890, A121539, A131323, A036554, A010060, A079523, A081706.

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 *)
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 *)

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

Extended by R. J. Mathar, Aug 28 2009

A161673 Positions n such that A010060(n) + A010060(n+8) = 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103

Offset: 1

Vladimir Shevelev, Jun 16 2009

nonn

Also union of numbers of the form 8*A121539(n)+k, 0<=k<8.

Generalization: the numbers n such that A010060(n)+A010060(n+2^m)=1 constitute the union of sequences {2^m*A121539(n)+k}, k=0,1,...,2^m-1.

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

Cf. A161639, A161641, A161627, A161579, A161580, A121539, A079523, A131323, A036554, A010060, A121539, A079523, A081706.

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 + 8] == 1, Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

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

A001477 \ A161639.

Edited and extended by R. J. Mathar, Sep 02 2009

A161817 Positions n such that A010060(n) = A010060(n+5).

Original entry on oeis.org

0, 2, 5, 8, 10, 11, 12, 14, 15, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 43, 44, 46, 47, 48, 50, 53, 56, 58, 59, 60, 62, 63, 64, 66, 69, 72, 74, 75, 76, 78, 79, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 107, 108, 110, 111, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 139, 140, 142, 143, 144

Offset: 1

Vladimir Shevelev, Jun 20 2009

Let A=Axxxxxx be any sequence. Denote by A^* the intersection of A and the union of sequences {4*A(n)+k}, k=-1,0,1,2. Then the present sequence is the union of A079523^* and A121539^*.

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]

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.

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

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 + 5], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n+5)==Mod(hammingweight(n),2) \\ Charles R Greathouse IV, Mar 26 2013

A161824 Numbers such that A010060(n) = A010060(n+6).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 44, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 61, 64, 65, 66, 67, 70, 71, 72, 73, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 90, 91, 94, 95, 96, 97, 98, 99, 102, 103, 104, 105, 108

Offset: 1

Vladimir Shevelev, Jun 20 2009

Let A=Axxxxxx be any sequence from OEIS. Denote by A^* the intersection of the union of sequences {2*A(n)+j}, j=0,1, and the union of sequences {4*A(n)+k}, k=-2,-1,0,1. Then the sequence is the union of (A079523)^* and (A121539)^*.

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

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 + 6], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)

is(n)=hammingweight(n)%2==hammingweight(n+6)%2 \\ Charles R Greathouse IV, Aug 20 2013

Terms a(40) onwards added by G. C. Greubel, Jan 05 2018

Offset corrected by Mohammed Yaseen, Mar 29 2023

A161890 Numbers such that A010060(n) = A010060(n+9).

Original entry on oeis.org

0, 2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 19, 20, 22, 24, 26, 27, 28, 30, 32, 34, 35, 36, 38, 39, 41, 45, 47, 48, 50, 51, 52, 54, 55, 57, 61, 63, 64, 66, 67, 68, 70, 71, 73, 77, 79, 80, 82, 83, 84, 86, 88, 90, 91, 92, 94, 96, 98, 99, 100, 102, 103, 105, 109, 111, 112, 114, 115, 116, 118, 120

Offset: 0

Vladimir Shevelev, Jun 21 2009

Or union of intersection of A161639 and {A079523(n)-8} and intersection of A161673 and {A121539(n)-8}. In general, for a>=1, consider equations A010060(x+a)+A010060(x)=1, A010060(x+a)=A010060(x). Denote via B_a (C_a) the sequence of nonnegative solutions of the first (second) equation. Then we have recursions: B_(a+1) is the union of transactions 1) C_a and {A121539(n)-a}, 2) B_a and {A079523(n)-a}; C_(a+1) is the union of transactions 1) C_a and {A079523(n)-a}, 2) B_a and {A121539(n)-a}.
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
This conjecture was actually proved in a later version of the Shevelev arxiv article cited below, and it can also easily be proved by the Walnut prover. - Jeffrey Shallit, Oct 12 2022

G. C. Greubel, Table of n, a(n) for n = 0..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.
Vladimir 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. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.

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 <= 18000, n++, If[tm[n] == tm[n + 9], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)
SequencePosition[ThueMorse[Range[0,150]],{x_,,,_,,,_,,,x_}][[All,1]]-1 (* Harvey P. Dale, Feb 06 2023 *)

is(n)=hammingweight(n)%2==hammingweight(n+9)%2 \\ Charles R Greathouse IV, Aug 20 2013

Terms a(35) onward added by G. C. Greubel, Jan 05 2018

A161916 The smallest k such that A010060(n+k)=A010060(n).

Original entry on oeis.org

3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 1

Offset: 0

Vladimir Shevelev, Jun 22 2009

nonn

Each entry is 1, 2 or 3, associated with positions registered in A079523, A081706, and A161579, respectively.

Cf. A010060, A079523, A081706, A161879

Definition shortened, sequence extended by R. J. Mathar, Nov 02 2010

cp's OEIS Frontend

A161580 Positions n such that A010060(n) + A010060(n+3) = 1.

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

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

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A161641 Positions n such that A010060(n) + A010060(n+4) = 1.

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A161674 Positions n such that A010060(n) + A010060(n+2) = 1.

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A161673 Positions n such that A010060(n) + A010060(n+8) = 1.

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A161817 Positions n such that A010060(n) = A010060(n+5).

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