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

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

Cf. A260390, A260453.

Sequence in context: A243556 A048233 A287730 * A005679 A232927 A350651

Adjacent sequences: A260449 A260450 A260451 * A260453 A260454 A260455

Keyword

nonn,easy

Author

Clark Kimberling, Aug 24 2015

Status

approved