A252703 Number of strings of length n over a 10-letter alphabet that do not begin with a palindrome.

0, 10, 90, 810, 8010, 79290, 792090, 7912890, 79120890, 791129610, 7911216810, 79111376010, 791112968010, 7911121767210, 79111209759210, 791112018471210, 7911120105591210, 79111200264782490, 791112001856695290, 7911120010655736090, 79111200098646144090

Offset: 0

Author

Peter Kagey, Dec 20 2014

Keywords

• easy
• nonn
• walk

Comments

10 divides a(n) for all n.

lim n -> infinity a(n)/10^n ~ 0.79111200088977 is the probability that a random, infinite string over a 10-letter alphabet does not begin with a palindrome.

This sequence gives the number of walks on K_10 with loops that do not begin with a palindromic sequence.

Examples

			For n = 3, the first 20 of the a(3) = 810 solutions are (in lexicographic order) 011, 012, 013, 014, 015, 016, 017, 018, 019, 021, 022, 023, 024, 025, 026, 027, 028, 029, 031, 032.
		

Links

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

Crossrefs

A249643 gives the number of strings of length n over a 10-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), A252702 (k=9).

Programs

Mathematica

a252703[n_] := Block[{f},
  f[0] = f[1] = 0;
  f[x_] := 10*f[x - 1] + 10^Ceiling[(x)/2] - f[Ceiling[(x)/2]];
Prepend[Rest@Table[10^i - f[i], {i, 0, n}],0]]; a252703[20] (* Michael De Vlieger, Dec 26 2014 *)

Ruby

seq = [1, 0]; (2..N).each { |i| seq << 10 * seq[i-1] + 10**((i+1)/2) - seq[(i+1)/2] }; seq = seq.each_with_index.collect { |a, i| 10**i - a }

Formula

a(n) = 10^n - A249643(n) for n > 0.