[seqfan] Re: Remark relating to A175096 and A006995

[franktaw at netscape.net]

Any permutation of the runs of n gives another such permutation when 
reversed. This pairs up all non-palindromic permutations of the runs of 
n. Thus the parity of A175096(n) is the parity of the number of 
palindromic run-permutations of n. For small n, this is 1 when n is a 
palindrome, and 0 otherwise.

The first exception is 43, binary 101011, which has a non-trivial 
palindromic run-permutation 45, binary 101101.

Another kind of exception occurs first for n = 365, binary 101101101, 
which is a palindrome, but has  another palindromic run-permutation 
427, binary 110101011.

Franklin T. Adams-Watters

-----Original Message-----
From: Jeremy Gardiner <jeremy.gardiner at btinternet.com>

I am curious about the similarity between the two sequences below:

It appears that A175096(n) is *almost always* odd if n is in A006995 and
even otherwise, but I don't understand A175096 well enough to 
understand why
this should be?

a(n) = parity of A175096
b(n) = characteristic function of A006995 (binary palindromes)

a(n)
,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,
0,0
,0,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,
0,0
,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,
b(n)
1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0
,0,
0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0
,0,
0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0
,0,
0,0,0,0,0,1,0,0,0,0,0,0,0,1

Here is the description of A175096:

Write n in binary (without leading 0's). A175096(n) = the number of 
distinct
numerical values made by permutating the runs of 0's and the runs of 
1's,
such that the runs (of nonzero length) of 1's alternate with the runs 
(of
nonzero length) of 0's. The permutated binary numbers (those not equal 
to n)
may start with leading 0's.

Regards,
Jeremy Gardiner




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