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
Formula: a(n) = 1 - A259599(n).
EXAMPLE
w(1) = 010, the initial word.
w(2) = 0100010 ( = 010+0+010, where + = concatenation)
w(3) = w(2)+1+w(2)*
w(4) = w(3)+0+w(3)*
MATHEMATICA
u[1] = { 0, 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
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 22 2015
STATUS
approved