

A254128


Number of binary strings of length n that begin with an oddlength palindrome.


3



0, 0, 0, 4, 8, 20, 40, 88, 176, 364, 728, 1480, 2960, 5960, 11920, 23920, 47840, 95828, 191656, 383608, 767216, 1535000, 3070000, 6141136, 12282272, 24566776, 49133552, 98271568, 196543136, 393095120, 786190240, 1572398176, 3144796352, 6289627948, 12579255896
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OFFSET

0,4


COMMENTS

This sequence gives the number of binary strings of length n that begin with an oddlength palindrome (not including the trivial palindrome of length one).
'1011' is an example of a string that begins with an oddlength palindrome: the palindrome '101', which is of length 3.
'1101' is an example of a string that does not begin with an oddlength palindrome. (It does begin with the evenlength palindrome '11'.)
The probability of a random infinite binary string beginning with an oddlength palindrome is given by: limit n > infinity a(n)/(2^n), which is approximately 0.7322131597821109.


LINKS

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


FORMULA

a(2*n) = 2*a(2*n1) = A094536(2*n)  A003000(n) for all n > 0.
a(2*n+1) = 2*a(2*n) + 2^(n+1)  a(n+1) = A094536(2*n+1) for all n.


EXAMPLE

For n = 4 the a(3) = 8 solutions are: 0000 0001 0100 0101 1010 1011 1110 1111.


PROG

(Ruby)
s = [0, 0]
(2..N).each { n s << 2 * s[1] + (n.even? ? 0 : 2**(n/2+1)  s[n/2+1]) }


CROSSREFS

Cf. A003000. A094536 is the analogous sequence for evenlength palindromes.
Sequence in context: A156303 A301138 A008136 * A047196 A009889 A334706
Adjacent sequences: A254125 A254126 A254127 * A254129 A254130 A254131


KEYWORD

nonn,base,easy


AUTHOR

Peter Kagey, Jan 25 2015


STATUS

approved



