login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A249629 Number of strings of length n over a 4-letter alphabet that begin with a nontrivial palindrome. 9
0, 0, 4, 28, 124, 532, 2164, 8788, 35284, 141628, 567004, 2269948, 9081724, 36334492, 145345564, 581412508, 2325680284, 9302841652, 37211487124, 148846430068, 595386201844, 2381546731732, 9526188851284, 38104763100628, 152419060098004, 609676271166388, 2438705115439924, 9754820584849588 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A nontrivial palindrome is a palindrome of length 2 or greater. (I.e., "1" is a trivial palindrome, but "11" and "121" are nontrivial palindromes.)

For example, 0032 is a is a string of length 4 over a 4-letter alphabet that begins with a nontrivial palindrome (00).

4 divides a(n) for all n.

Number of walks of n steps that begin with a palindromic sequence on the complete graph K_4 with loops. (E.g., 0, 1, 1, 0, 3, 1, 2 is a valid walk with 7 steps and begins with the palindromic sequence '0110'.)

lim n -> infinity a(n)/4^n ~ 0.5415013252744246 is the probability that a random, infinite base-4 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) = 4*a(n) + 4^ceiling((n+1)/2) - a(ceiling((n+1)/2)).

EXAMPLE

for n=3 the a(3) = 28 solutions are: 000, 001, 002, 003, 010, 020, 030, 101, 110, 111, 112, 113, 121, 131, 202, 212, 220, 221, 222, 223, 232, 303, 313, 323, 330, 331, 332, 333.

MATHEMATICA

a249629[n_] := Block[{f},

  f[0] = f[1] = 0;

  f[x_] := 4*f[x - 1] + 4^Ceiling[x/2] - f[Ceiling[x/2]];

Table[f[i], {i, 0, n}]]; a249629[27] (* Michael De Vlieger, Dec 27 2014 *)

PROG

(Ruby) seq = [0, 0]; (2..N).each{ |i| seq << 4 * seq[i-1] + 4**((i+1)/2) - seq[(i+1)/2] }

(Haskell)

import Data.Ratio

a 0 = 0; a 1 = 0;

a n = 4 * a(n - 1) + 4^ceiling(n % 2) - a(ceiling(n % 2)) -- Peter Kagey, Aug 13 2015

(MAGMA) [0] cat  [n le 1 select 0 else 4*Self(n-1) + 4^Ceiling((n)/2) - Self(Ceiling((n)/2)): n in [1..40]]; // Vincenzo Librandi, Aug 20 2015

CROSSREFS

Analogous sequences for k-letter alphabets: A248122 (k=3), A249638 (k=5), A249639 (k=6), A249640 (k=7), A249641 (k=8), A249642 (k=9), A249643 (k=10).

Sequence in context: A318011 A212900 A196514 * A131459 A231581 A223115

Adjacent sequences:  A249626 A249627 A249628 * A249630 A249631 A249632

KEYWORD

easy,nonn,walk

AUTHOR

Peter Kagey, Nov 02 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 20 09:27 EDT 2019. Contains 324234 sequences. (Running on oeis4.)