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%I #36 Sep 08 2022 08:46:10
%S 0,0,4,28,124,532,2164,8788,35284,141628,567004,2269948,9081724,
%T 36334492,145345564,581412508,2325680284,9302841652,37211487124,
%U 148846430068,595386201844,2381546731732,9526188851284,38104763100628,152419060098004,609676271166388,2438705115439924,9754820584849588
%N Number of strings of length n over a 4-letter alphabet that begin with a nontrivial palindrome.
%C 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.)
%C For example, 0032 is a is a string of length 4 over a 4-letter alphabet that begins with a nontrivial palindrome (00).
%C 4 divides a(n) for all n.
%C 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'.)
%C lim n -> infinity a(n)/4^n ~ 0.5415013252744246 is the probability that a random, infinite base-4 string begins with a nontrivial palindrome.
%H Peter Kagey, <a href="/A249629/b249629.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0) = 0; a(1) = 0; a(n+1) = 4*a(n) + 4^ceiling((n+1)/2) - a(ceiling((n+1)/2)).
%e 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.
%t a249629[n_] := Block[{f},
%t f[0] = f[1] = 0;
%t f[x_] := 4*f[x - 1] + 4^Ceiling[x/2] - f[Ceiling[x/2]];
%t Table[f[i], {i, 0, n}]]; a249629[27] (* _Michael De Vlieger_, Dec 27 2014 *)
%o (Ruby) seq = [0, 0]; (2..N).each{ |i| seq << 4 * seq[i-1] + 4**((i+1)/2) - seq[(i+1)/2] }
%o (Haskell)
%o import Data.Ratio
%o a 0 = 0; a 1 = 0;
%o a n = 4 * a(n - 1) + 4^ceiling(n % 2) - a(ceiling(n % 2)) -- _Peter Kagey_, Aug 13 2015
%o (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
%Y 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).
%K easy,nonn,walk
%O 0,3
%A _Peter Kagey_, Nov 02 2014
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