A248122 Number of strings of length n over a three-letter alphabet that begin with a nontrivial palindrome.

0, 0, 3, 15, 51, 165, 507, 1551, 4683, 14127, 42459, 127599, 383019, 1149693, 3449715, 10351023, 31054947, 93170397, 279516747, 838566831, 2515717083, 7547200797, 22641651939, 67925104239, 203775461139, 611326828047, 1833980928771

Offset: 0

Author

Peter Kagey, Oct 28 2014

Keywords

• easy
• nonn
• walk

Comments

A nontrivial palindrome is a palindrome of length two or greater. (I.e., "1" is a trivial palindrome, but "11" and "121" are nontrivial palindromes.)

For example, 0012 is a string of length four over a three-letter alphabet that begins with a nontrivial palindrome (00).

3 divides a(n) for all n: 0, 0, 1, 5, 17, 55, 169, 517, 1561, 4709, 14153, ...

Number of walks of n steps that begin with a palindromic sequence on the complete graph K_3 with loops. (E.g., 0, 1, 1, 0, 2, 1, 2 is a valid walk with 7 steps and begins with the palindromic sequence '0110'.)

lim n -> infinity a(n)/3^n ~ 0.721510080117 is the probability that a random, infinite base-3 string begins with a nontrivial palindrome.

Examples

			For n = 3, the a(3) = 15 solutions are 000, 001, 002, 010, 020, 101, 110, 111, 112, 121, 202, 212, 220, 221, 222.
		

Links

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

Crossrefs

Analogous sequences for k-letter alphabets: A249629 (k=4), A249638 (k=5), A249639 (k=6), A249640 (k=7), A249641 (k=8), A249642 (k=9), A249643 (k=10).

Programs

Haskell

import Data.Ratio
a 0 = 0; a 1 = 0;
a n = 3 * a(n - 1) + 3^ceiling(n % 2) - a(ceiling(n % 2)) -- Peter Kagey, Aug 13 2015

Mathematica

a248122[n_] := Block[{f},
  f[0] = f[1] = 0;
  f[x_] := 3*f[x - 1] + 3^Ceiling[x/2] - f[Ceiling[x/2]];
Table[f[i], {i, 0, n}]]; a248122[26] (* Michael De Vlieger, Dec 27 2014 *)

Ruby

seq = [0, 0]; (2..N).each{ |i| seq << 3 * seq[i-1] + 3**((i+1)/2) - seq[(i+1)/2] }

Formula

a(0) = 0; a(1) = 0; a(n) = 3*a(n-1) + 3^ceiling(n/2) - a(ceiling(n/2)), for n >= 2.