



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
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OFFSET

0,2


COMMENTS

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


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000
J.P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the ThueMorse sequence, arXiv:1401.3727 [math.NT], 2014.
J.P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the ThueMorse sequence, J. de ThÃ©orie des Nombres de Bordeaux, 27, no. 2 (2015), 375388.
V. Shevelev,Equations of the form t(x+a)=t(x) and t(x+a)=1t(x) for ThueMorse sequence, arXiv:0907.0880[math.NT], 20092012.


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 <= 18000, n++, If[tm[n] == tm[n + 9], Sow[n]]]][[2, 1]] (* G. C. Greubel, Jan 05 2018 *)


PROG

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


CROSSREFS

Cf. A161824, A161817, A161674, A161673, A161639, A161641, A161627, A161579, A161580, A121539, A131323, A036554, A010060, A079523, A081706.
Sequence in context: A050050 A222801 A117307 * A089388 A055494 A239115
Adjacent sequences: A161887 A161888 A161889 * A161891 A161892 A161893


KEYWORD

nonn,base


AUTHOR

Vladimir Shevelev, Jun 21 2009


EXTENSIONS

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


STATUS

approved



