A249640 Number of strings of length n over a 7-letter alphabet that begin with a nontrivial palindrome.

0, 0, 7, 91, 679, 5005, 35287, 248731, 1742839, 12211675, 85493527, 598537051, 4189841719, 29329466845, 205306842727, 1437151921051, 10060067469319, 70420500427165, 492943531132087, 3450604914906331, 24154234601326039, 169079643588071965

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, 0062 is a string of length 4 over a seven letter alphabet that begins with a nontrivial palindrome (00).

7 divides a(n) for all n.

Number of walks of n steps that begin with a palindromic sequence on the complete graph K_7 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'.)

lim n -> infinity a(n)/7^n ~ 0.30271398450898696 is the probability that a random, infinite base-7 string 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) = 7*a(n) + 7^ceiling((n+1)/2) - a(ceiling((n+1)/2)).

Example

For n=3 the a(3) = 91 solutions are: 000, 001, 002, 003, 004, 005, 006, 010, 020, 030, 040, 050, 060, 101, 110, 111, 112, 113, 114, 115, 116, 121, 131, 141, 151, 161, 202, 212, 220, 221, 222, 223, 224, 225, 226, 232, 242, 252, 262, 303, 313, 323, 330, 331, 332, 333, 334, 335, 336, 343, 353, 363, 404, 414, 424, 434, 440, 441, 442, 443, 444, 445, 446, 454, 464, 505, 515, 525, 535, 545, 550, 551, 552, 553, 554, 555, 556, 565, 606, 616, 626, 636, 646, 656, 660, 661, 662, 663, 664, 665, 666

Mathematica

a249640[n_] := Block[{f},
  f[0] = f[1] = 0;
  f[x_] := 7*f[x - 1] + 7^Ceiling[x/2] - f[Ceiling[x/2]];
Table[f[i], {i, 0, n}]]; a249640[21] (* Michael De Vlieger, Dec 27 2014 *)

Prog

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

Crossrefs

Analogous sequences for k-letter alphabets: A248122 (k=3), A249629 (k=4), A249638 (k=5), A249639 (k=6), A249641 (k=8), A249642 (k=9), A249643(k=10).

Sequence in context: A221132 A181475 A374094 * A248226 A165230 A346939

Adjacent sequences: A249637 A249638 A249639 * A249641 A249642 A249643

Keyword

easy,nonn,walk

Author

Peter Kagey, Nov 02 2014

Status

approved