A252701 Number of strings of length n over an 8-letter alphabet that do not begin with a palindrome.

0, 8, 56, 392, 3080, 24248, 193592, 1545656, 12362168, 98873096, 790960520, 6327490568, 50619730952, 404956301960, 3239648870024, 25917178598024, 207337416422024, 1658699232503096, 13269593761151672, 106156749298252856, 849253993595062328, 6794031942433008056

Offset: 0

Author

Peter Kagey, Dec 20 2014

Keywords

• easy
• nonn
• walk

Comments

8 divides a(n) for all n.

lim n -> infinity a(n)/8^n ~ 0.73661041899617 is the probability that a random, infinite string over an 8-letter alphabet does not begin with a palindrome.

This sequence gives the number of walks on K_8 with loops that do not begin with a palindromic sequence.

Examples

			For n = 3, the first 10 of the a(3) = 392 solutions are (in lexicographic order) 011, 012, 013, 014, 015, 016, 017, 021, 022, 023.
		

Links

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

Crossrefs

A249641 gives the number of strings of length n over an 8-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), A252702 (k=9), A252703 (k=10).

Programs

Mathematica

a252701[n_] := Block[{f}, f[0] = f[1] = 0;
  f[x_] := 8*f[x - 1] + 8^Ceiling[(x)/2] - f[Ceiling[(x)/2]];
Prepend[Rest@Table[8^i - f[i], {i, 0, n}], 0]]; a252701[21] (* Michael De Vlieger, Dec 26 2014 *)

Ruby

seq = [1, 0]; (2..N).each { |i| seq << 8 * seq[i-1] + 8**((i+1)/2) - seq[(i+1)/2] }; seq = seq.each_with_index.collect { |a, i| 8**i - a }

Formula

a(n) = 8^n - A249641(n) for n > 0.