cp's OEIS Frontend

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.

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

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

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.

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]

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)^*.

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

One can prove that if in the sequence of numbers N for which A010060(N+2^n)= A010060(N) you replace the odious (evil) terms by 1's (0's), then we obtain 2^(n+1)-periodic (0,1)-sequence; if you write it in the form .xx...,i.e., as a binary infinite fraction, then the corresponding fraction has the form a(n)/A000215(n). These fractions very fast converge to the Thue-Morse constant .4124540336401...; e.g a(5)/(2^32+1) approximates this constant up to 10^(-9). These approximations differ from A074072-A074073. Conjecture. For n>=1, the fraction a(n)/A000215(n) is a convergent corresponding to the continued fraction for the Thue-Morse constant.

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

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