A260397 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,1,0) and midword sequence (a(n)); see Comments.

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

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.

Links

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

Formula

a(n) = 1 - A260445(n).

Example

w(1) = 110, the initial word;
w(2) = w(1)+a(1)+w(1)* = 1101011 ( = 110+1+011, where + = concatenation);
w(3) = w(2)+a(2)+w(2)* = 110101111101011 (= 1101011+1+1101011);
w(4) = w(3)+a(3)+w(3)* = w(3)+0+w(3)*.

Mathematica

u[1] = {1, 1, 0}; 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, A260445, A260394.

Sequence in context: A131372 A098457 A284793 * A137161 A329677 A077050

Adjacent sequences: A260394 A260395 A260396 * A260398 A260399 A260400

Keyword

nonn,easy

Author

Clark Kimberling, Aug 13 2015

Status

approved