A260452 - OEIS

A260452

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

2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 2, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1

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OFFSET

1,1

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

EXAMPLE

w(1) = 213, the initial word.

w(2) = 2132312 ( = 213+2+312, where + = concatenation)

w(3) = w(2)+1+w(2)*

w(4) = w(3)+3+w(3)*

MATHEMATICA

u[1] = {2, 1, 3}; 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]]}; v = u[8]

CROSSREFS