# [seqfan] Remark relating to A175096 and A006995

Jeremy Gardiner jeremy.gardiner at btinternet.com
Thu May 20 18:41:06 CEST 2010

```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)