A260390 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,0) and midword sequence (a(n)); see Comments.

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 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. Examples follow:

initial word midword sequence inf. palindr. word |w(n)|

w(1) = 10 m(i) = a(i) A260390 A083329

w(1) = 01 m(i) = a(i) A260393 A083329

w(1) = 011 m(i) = a(i) A260394 A000225

w(1) = 110 m(i) = a(i) A260397 A000225

w(1) = 101 m(i) = a(i) A035263 A000225

w(1) = 100 m(i) = a(i) A260444 A000225

w(1) = 001 m(i) = a(i) A260445 A000225

w(1) = 010 m(i) = a(i) A260446 A000225

w(1) = 0 m(i) = i A007814 A000225

w(1) = 123 m(i) = a(i) A260449 A000225

w(1) = 132 m(i) = a(i) A260450 A000225

w(1) = 231 m(i) = a(i) A260451 A000225

w(1) = 213 m(i) = a(i) A260452 A000225

w(1) = 321 m(i) = a(i) A260453 A000225

w(1) = 312 m(i) = a(i) A260454 A000225

w(1) = 0 (see A260455) A260455 A081254 (conjectured)

w(1) = 1 (see A260456) A260456 A081254 (conjectured)

As a sort of (obvious) converse of the above method for constructing infinite palindromic words, every such word is determined by an initial segment w(1) and a midword sequence (m(n)), where terms of the latter may be the empty word.

Links

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

Formula

a(n) = 1 - A260393(n).

Example

w(1) = 10, the initial word.
w(2) = 10101 ( = 10+1+01, where + = concatenation)
w(3) = 10101010101 = w(2)+0+w(2)*
w(4) = w(3)+1+w(3)*

Mathematica

u[1] = {1, 0}; m[1] = {u[1][[1]]};
u[n_] := u[n] = Join[u[n - 1], m[n - 1], Reverse[u[n - 1]]];
Table[Length[u[n]], {n, 1, 20}]  (* A083329 *)
Flatten[Position[u[8], 0]]   (* A260391 *)
Flatten[Position[u[8], 1]]   (* A260392 *)

Crossrefs

Cf. A083329, A260392, A260394.

Sequence in context: A250299 A361122 A193497 * A179758 A287795 A267704

Adjacent sequences: A260387 A260388 A260389 * A260391 A260392 A260393

Keyword

nonn,easy

Author

Clark Kimberling, Jul 31 2015

Status

approved