A259599 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,0,1) and midword sequence (a(n)); see Comments.

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

Offset

1

Comments

Below, w* denotes the reversal of a word w, and "sequence" and "word" are interchangable. An infinite word is palindromic if it has infinitely many initial subwords w such that w = w*.

Many infinite palindromic words (a(1),a(2),...) are determined by an initial word w and a midword sequence (m(1),m(2),...) of palindromes, as follows: for given w of length k, take w(1) = w = (a(1),a(2),...,a(k)). Form the palindrome w(2) = w(1)m(1)w(1)* by concatenating w(1), m(1), and w(1)*. Continue inductively; i.e., w(n+1) = w(n)m(n)w(n)* for all n >= 1. See A260390 for examples.

a(n) = A035263(n) for n=1..31, but not n=32.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Formula

a(n) = 1 - A260446(n).

Example

w(1) = 101, the initial word;
w(2) = 1011101 ( = 101+1+101, where + = concatenation);
w(3) = w(2)+0+w(2)* = 101110101011101;
w(4) = w(3)+1+w(3)*.

Mathematica

u[1] = {1, 0, 1}; m[1] = {u[1][[1]]};
u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
m[k_] := {u[k][[k]]}; u[6]

Crossrefs

Cf. A260390, A260446.

Sequence in context: A029883 A035263 A089045 * A070749 A285341 A059778

Adjacent sequences: A259596 A259597 A259598 * A259600 A259601 A259602

Keyword

nonn,easy

Author

Clark Kimberling, Aug 13 2015

Status

approved