A260450 Infinite palindromic word (a(1),a(2),a(3),...) with initial word w(1) = (1,3,2) and midword sequence (a(n)); see Comments.

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

Offset

1,2

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) = 132, the initial word.
w(2) = 1321231 ( = 132+1+231, where + = concatenation)
w(3) = w(2)+2+w(2)*
w(4) = w(3)+3+w(3)*

Mathematica

u[1] = {1, 3, 2}; 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, A260449.

Sequence in context: A068929 A060567 A174543 * A036583 A351557 A047878

Adjacent sequences: A260447 A260448 A260449 * A260451 A260452 A260453

Keyword

nonn,easy

Author

Clark Kimberling, Aug 24 2015

Status

approved