

A259599


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


4



1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
a(n) = A035263(n) for n=1..31, but not n=32.


LINKS

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


FORMULA

a(n) = 1  A260446(n).


EXAMPLE

w(1) = 101, the initial word;
w(2) = 1011101 ( = 101+1+101, where + = concatenation);
w(3) = w(2)+0+w(2)* = 101110101011101;
w(4) = w(3)+1+w(3)*.


MATHEMATICA

u[1] = {1, 0, 1}; 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

Cf. A260390, A260446.
Sequence in context: A029883 A035263 A089045 * A070749 A285341 A059778
Adjacent sequences: A259596 A259597 A259598 * A259600 A259601 A259602


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Aug 13 2015


STATUS

approved



