A252702 Number of strings of length n over a 9-letter alphabet that do not begin with a palindrome.

0, 9, 72, 576, 5112, 45432, 408312, 3669696, 33022152, 297153936, 2674339992, 24068651616, 216617456232, 1949553436392, 17545977257832, 157913762298336, 1421223827662872, 12791014151811912, 115119127069153272, 1036072140948039456, 9324649265858015112

Offset

0,2

Comments

9 divides a(n) for all n.

lim n -> infinity a(n)/9^n ~ 0.766976957370438 is the probability that a random, infinite string over a 9-letter alphabet does not begin with a palindrome.

This sequence gives the number of walks on K_9 with loops that do not begin with a palindromic sequence.

Links

Peter Kagey, Table of n, a(n) for n = 0..1000

Formula

a(n) = 9^n - A249642(n) for n > 0.

Example

For n = 3, the first 10 of the a(3) = 576 solutions are (in lexicographic order) 011, 012, 013, 014, 015, 016, 017, 018, 021, 022.

Prog

(Ruby) seq = [1, 0]; (2..N).each { |i| seq << 9 * seq[i-1] + 9**((i+1)/2) - seq[(i+1)/2] }; seq = seq.each_with_index.collect { |a, i| 9**i - a }

Crossrefs

A249642 gives the number of strings of length n over a 9-letter alphabet that DO begin with a palindrome.

Analogous sequences for k-letter alphabets: A252696 (k=3), A252697 (k=4), A252698 (k=5), A252699 (k=6), A252700 (k=7), A252701 (k=8), A252703 (k=10).

Sequence in context: A170642 A170690 A003951 * A033135 A127053 A001809

Adjacent sequences: A252699 A252700 A252701 * A252703 A252704 A252705

Keyword

easy,nonn,walk

Author

Peter Kagey, Dec 20 2014

Status

approved