A081706 -id:A081706 - cp's OEIS frontend

A079523 Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.

1, 5, 7, 9, 13, 17, 21, 23, 25, 29, 31, 33, 37, 39, 41, 45, 49, 53, 55, 57, 61, 65, 69, 71, 73, 77, 81, 85, 87, 89, 93, 95, 97, 101, 103, 105, 109, 113, 117, 119, 121, 125, 127, 129, 133, 135, 137, 141, 145, 149, 151, 153, 157, 159, 161, 165, 167, 169, 173, 177, 181

Offset: 1

Benoit Cloitre, Jan 21 2003

Also, n such that A010060(n) = A010060(n+1) where A010060 is the Thue-Morse sequence.
Sequence of n such that a(n) = 3n begins 7, 23, 27, 29, 31, 39, 71, 87, 91, 93, 95, ...
Values of k such that the Motzkin number A001006(2k) is even. Values of k such that the number of restricted hexagonal polyominoes with 2k+1 cells is even (see A002212). Values of k such that the number of directed animals of size k+1 is even (see A005773). Values of k such that the Riordan number A005043(k) is even. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
a(n) = A036554(n)-1 = A072939(n)-2. - Ralf Stephan, Jun 09 2003
Odious and evil terms alternate. - Vladimir Shevelev, Jun 22 2009
The sequence has the following fractal property: remove terms of the form 4k+1 from the sequence, and the remaining terms are of the form 4k+3: 7, 23, 31, 39, 55, 71, 87, ...; then subtract 3 from each of these terms and divide by 4 and you get the original sequence: 1, 5, 7, 9, 13, ... - Benoit Cloitre, Apr 06 2010
A035263(a(n)) = 0. - Reinhard Zumkeller, Mar 01 2012

Reinhard Zumkeller, 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 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.
J.-P. Allouche, A. Arnold, J. Berstel, S. Brlek, W. Jockusch, Simon Plouffe, and B. E. Sagan, A relative of the Thue-Morse sequence, Discrete Math., 139, 1995, 455-461.
Narad Rampersad and Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
Thomas Zaslavsky, Anti-Fibonacci Numbers: A Formula, Sep 26 2016 [Introduces the name "utterly odd". - _N. J. A. Sloane_, Sep 27 2016]
Index entries for 2-automatic sequences.

Cf. A003159, A003157, A003158, A075326, A131323.

import Data.List (elemIndices)
a079523 n = a079523_list !! (n-1)
a079523_list = elemIndices 0 a035263_list
-- Reinhard Zumkeller, Mar 01 2012

[n: n in [0..200] | Valuation(n+1, 2) mod 2 eq 0 + 1]; // Vincenzo Librandi, Apr 16 2015

Select[ Range[200], MatchQ[ IntegerDigits[#, 2], {b : (1) ..} | {_, 0, b : (1) ..} /; OddQ[ Length[{b}]]] & ] (* Jean-François Alcover, Jun 17 2013 *)

is(n)=valuation(n+1,2)%2 \\ Charles R Greathouse IV, Mar 07 2013

from itertools import count, islice
def A079523_gen(startvalue=1): return filter(lambda n:(~(n+1)&n).bit_length()&1,count(max(startvalue,1))) # generator of terms >= startvalue
A079523_list = list(islice(A079523_gen(),30)) # Chai Wah Wu, Jul 05 2022

def A079523(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n)-1 # Chai Wah Wu, Jan 29 2025

a(n) is asymptotic to 3n.
a(n) = 2*A003159(n) - 1. a(1)=1, a(n) = a(n-1) + 2 if (a(n-1)+1)/2 does not belong to the sequence and a(n) = a(n-1) + 4 otherwise. - Emeric Deutsch and Bruce E. Sagan, Apr 02 2003
a(n) = (1/2)*A081706(2n-1).
a(n) = A003158(n) - n = A003157(n) - n - 1. - Philippe Deléham, Feb 22 2004
Values of k such that A091297(k) = 0. - Philippe Deléham, Feb 25 2004

A007413 A squarefree (or Thue-Morse) ternary sequence: closed under 1->123, 2->13, 3->2. Start with 1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

a(n)=2 if and only if n-1 is in A079523. - Benoit Cloitre, Mar 10 2003
Partial sums modulo 4 of the sequence 1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), a(6), ... - Philippe Deléham, Mar 04 2004
To construct the sequence: start with 1 and concatenate 4 -1 = 3: 1, 3, then change the last term (2 -> 1, 3 ->2 ) gives 1, 2. Concatenate 1, 2 with 4 -1 = 3, 4 - 2 = 2: 1, 2, 3, 2 and change the last term: 1, 2, 3, 1. Concatenate 1, 2, 3, 1 with 4 - 1 = 3, 4 - 2 = 2, 4 - 3 = 1, 4 - 1 = 3: 1, 2, 3, 1, 3, 2, 1, 3 and change the last term: 1, 2, 3, 1, 3, 2, 1, 2 etc. - Philippe Deléham, Mar 04 2004
To construct the sequence: start with the Thue-Morse sequence A010060 = 0, 1, 1, 0, 1, 0, 0, 1, ... Then change 0 -> 1, 2, 3,  and 1 -> 3, 2, 1,  gives: 1, 2, 3, , 3, 2, 1, ,3, 2, 1, , 1, 2, 3, , 3, 2, 1, , ... and fill in the successive holes with the successive terms of the sequence itself. - _Philippe Deléham, Mar 04 2004
To construct the sequence: to insert the number 2 between the A003156(k)-th term and the (1 + A003156(k))-th term of the sequence 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ... - Philippe Deléham, Mar 04 2004
Conjecture. The sequence is formed by the numbers of 1's between every pair of consecutive 2's in A076826. - Vladimir Shevelev, May 31 2009

			Here are the first 5 stages in the construction of this sequence, together with Mma code, taken from Keranen's article. His alphabet is a,b,c rather than 1,2,3.
productions = {"a" -> "abc ", "b" -> "ac ", "c" -> "b ", " " -> ""};
NestList[g, "a", 5] // TableForm
a
abc
abc ac b
abc ac b abc b ac
abc ac b abc b ac abc ac b ac abc b
abc ac b abc b ac abc ac b ac abc b abc ac b abc b ac abc b abc ac b ac

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Thue, Über unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, No. 7 (1906), 1-22.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Roger L. Bagula, Description of sequence as successive rows of a triangle
James D. Currie, Non-repetitive words: Ages and essences, Combinatorica 16.1 (1996): 19-40. See p. 20.
James D. Currie, Palindrome positions in ternary square-free words, Theoretical Computer Science, 396 (2008) 254-257.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
V. Keranen, New Abelian Square-Free DT0L-Languages over 4 Letters, Theoretical Computer Science, Volume 410, Issues 38-40, 6 September 2009, Pages 3893-3900.
S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Andrzej Tomski and Maciej Zakarczemny, A note on Browkin's and Cao's cancellation algorithm, Technical Transections 7/2018.

Cf. A001285, A010060.
First differences of A000069.
Equals A036580(n-1) + 1.
Cf. A115384, A159481, A079523, A000120.

Nest[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {1, 3}, 3 -> {2}}] &, {1}, 7] (* Robert G. Wilson v, May 07 2005 *)
2 - Differences[ThueMorse[Range[0, 100]]] (* Paolo Xausa, Oct 25 2024 *)

{a(n) = if( n<1 || valuation(n, 2)%2, 2, 2 + (-1)^subst( Pol(binary(n)), x,1))};

def A007413(n): return 2-(n.bit_count()&1)+((n-1).bit_count()&1) # Chai Wah Wu, Mar 03 2023

a(n) modulo 2 = A035263(n). a(A036554(n)) = 2. a(A003159(n)) = 1 if n odd. a(A003159(n)) = 3 if n even. a(n) = A033485(n) mod 4. a(n) = 4 - A036585(n-1). - Philippe Deléham, Mar 04 2004
a(n) = 2 - A029883(n) = 3 - A036577(n). - Philippe Deléham, Mar 20 2004
For n>=1, we have: 1) a(A108269(n))=A010684(n-1); 2) a(A079523(n))=A010684(n-1); 3) a(A081706(2n))=A010684(n). - Vladimir Shevelev, Jun 22 2009

A039963 The period-doubling sequence A035263 repeated.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane

nonn

An example of a d-perfect sequence.
Motzkin numbers mod 2. - Benoit Cloitre, Mar 23 2004
Let {a, b, c, c, a, b, a, b, a, b, c, c, a, b, ...} be the fixed point of the morphism: a -> ab, b -> cc, c -> ab, starting from a; then the sequence is obtained by taking a = 1, b = 1, c = 0. - Philippe Deléham, Mar 28 2004
The asymptotic mean of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021
The Gilbreath transform of floor(log_2(n)) (A000523). - Thomas Scheuerle, Sep 02 2024

Seiichi Manyama, Table of n, a(n) for n = 0..10000
T. Amdeberhan and M. Alekseyev, A moment sequence and Motzkin numbers. Modular coincidence?, MathOverflow, 2021.
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link.
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; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.

Cf. A000523, A005802, A081706.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Flatten[ Nest[ Function[l, {Flatten[(l /. {a -> {a, b}, b -> {c, c}, c -> {a, b}})]}], {a}, 7] /. {a -> {1}, b -> {1}, c -> {0}}] (* Robert G. Wilson v, Feb 26 2005 *)

A039963(n) = 1 - valuation(n\2+1,2)%2; \\ Max Alekseyev, Oct 23 2021

def A039963(n): return ((m:=(n>>1)+1)&-m).bit_length()&1 # Chai Wah Wu, Jan 09 2023

a(n) = A035263(1+floor(n/2)). - Benoit Cloitre, Mar 23 2004
a(n) = A040039(n) mod 2 = A002212(n+1) mod 2. a(0) = a(1) = 1, for n>=2: a(n) = ( a(n) + Sum_{k=0..n-2} a(k)*a(n-2-k)) mod 2. - Philippe Deléham, Mar 26 2004
a(n) = (A(n+2) - A(n)) mod 2, for A = A019300, A001285, A010060, A010059, A000069, A001969. - Philippe Deléham, Mar 28 2004
a(n) = A001006(n) mod 2. - Christian G. Bower, Jun 12 2005
a(n) = (-1)^n*(A096268(n+1) - A096268(n)). - Johannes W. Meijer, Feb 02 2013
a(n) = 1 - A007814(floor(n/2)+1) mod 2 = A005802(n) mod 2. - Max Alekseyev, Oct 23 2021

More terms from Christian G. Bower, Jun 12 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe and Ralf Stephan, Jul 13 2007

A161579 Positions n such that A010060(n) = A010060(n+3).

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 38, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 54, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 70, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 105, 107

Offset: 1

Vladimir Shevelev, Jun 14 2009

Or: union of A131323 with the sequence of terms of the form A131323(n)-2, and with the sequence of terms of the form A036554(n)-2.

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. [_Vladimir Shevelev_, Jul 31 2009]

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

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

is(n)=hammingweight(n)%2==hammingweight(n+3)%2 \\ Charles R Greathouse IV, Aug 20 2013

Equals {A001477} \ {A161580}.

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A079523 Utterly odd numbers: numbers whose binary representation ends in an odd number of ones.

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A007413 A squarefree (or Thue-Morse) ternary sequence: closed under 1->123, 2->13, 3->2. Start with 1.

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A039963 The period-doubling sequence A035263 repeated.

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A161579 Positions n such that A010060(n) = A010060(n+3).

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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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