A252700 Number of strings of length n over a 7-letter alphabet that do not begin with a palindrome.

0, 7, 42, 252, 1722, 11802, 82362, 574812, 4021962, 28141932, 196981722, 1378789692, 9651445482, 67559543562, 472916230122, 3310409588892, 23172863100282, 162210013560042, 1135470066778362, 7948290270466812, 55638031696285962, 389466220495212042

Offset: 0

Author

Peter Kagey, Dec 20 2014

Keywords

• easy
• nonn
• walk

Comments

7 divides a(n) for all n.

lim n -> infinity a(n)/7^n ~ 0.697286015491013 is the probability that a random, infinite string over a 7-letter alphabet does not begin with a palindrome.

This sequence gives the number of walks on K_7 with loops that do not begin with a palindromic sequence.

Examples

			For n = 3, the first 10 of the a(3) = 252 solutions are (in lexicographic order) 011, 012, 013, 014, 015, 016, 021, 022, 023, 024.
		

Links

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

Crossrefs

A249640 gives the number of strings of length n over a 7-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), A252701 (k=8), A252702 (k=9), A252703 (k=10).

Programs

Mathematica

a252700[n_] := Block[{f}, f[0] = f[1] = 0;
  f[x_] := 7*f[x - 1] + 7^Ceiling[(x)/2] - f[Ceiling[(x)/2]];
Prepend[Rest@Table[7^i - f[i], {i, 0, n}], 0]]; a252700[21] (* Michael De Vlieger, Dec 26 2014 *)

Ruby

seq = [1, 0]; (2..N).each { |i| seq << 7 * seq[i-1] + 7**((i+1)/2) - seq[(i+1)/2] }; seq = seq.each_with_index.collect { |a, i| 7**i - a }

Formula

a(n) = 7^n - A249640(n) for n > 0.