OFFSET
0,3
COMMENTS
A nontrivial palindrome is a palindrome of length 2 or greater. (E.g., "1" is a trivial palindrome, but "11" and "121" are nontrivial palindromes.)
For example, 0042 is a string in a five letter alphabet of length 4 that begins with a nontrivial palindrome (00).
5 divides a(n) for all n.
Number of walks of n steps that begin with a palindromic sequence on the complete graph K_5 with loops. (E.g., 0, 1, 1, 0, 4, 1, 2 is a valid walk with 7 steps and begins with the palindromic sequence '0110'.)
Limit_{n->oo} a(n)/5^n ~ 0.429951613027098 is the probability that a random, infinite string in a five letter alphabet begins with a nontrivial palindrome.
LINKS
Peter Kagey, Table of n, a(n) for n = 0..1000
FORMULA
a(0) = 0; a(1) = 0; a(n+1) = 5*a(n) + 5^ceiling((n+1)/2) - a(ceiling((n+1)/2)).
EXAMPLE
For n=3 the a(3) = 45 valid strings are: 000, 001, 002, 003, 004, 010, 020, 030, 040, 101, 110, 111, 112, 113, 114, 121, 131, 141, 202, 212, 220, 221, 222, 223, 224, 232, 242, 303, 313, 323, 330, 331, 332, 333, 334, 343, 404, 414, 424, 434, 440, 441, 442, 443, 444.
MATHEMATICA
a249638[n_] := Block[{f},
f[0] = f[1] = 0;
f[x_] := 5*f[x - 1] + 5^Ceiling[x/2] - f[Ceiling[x/2]];
Table[f[i], {i, 0, n}]]; a249638[23] (* Michael De Vlieger, Dec 27 2014 *)
PROG
(Ruby) seq = [0, 0]; (2..N).each{ |i| seq << 5 * seq[i-1] + 5**((i+1)/2) - seq[(i+1)/2] }
(Haskell)
import Data.Ratio
a 0 = 0; a 1 = 0;
a n = 5 * a(n - 1) + 5^ceiling(n % 2) - a(ceiling(n % 2)) -- Peter Kagey, Aug 13 2015
CROSSREFS
KEYWORD
easy,nonn,walk
AUTHOR
Peter Kagey, Nov 02 2014
STATUS
approved