A014577

[Paul D. Hanna]

For sequences like these, the XOR BINOMIAL transform may reveal some
underlying structure.

(For definition of XOR BINOMIAL transform, please see:
http://www.research.att.com/projects/OEIS?Anum=A099884
http://www.research.att.com/projects/OEIS?Anum=A099887
http://www.research.att.com/projects/OEIS?Anum=A099888 )

The XOR BINOMIAL of A014577 is the sequence B:
B={1,
0,1,
1,0,1,0,
1,0,0,0,1,0,0,0,
1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,.
..}
 
Perhaps B can be used to find an interesting formula for A014577,
using: 

(*)  A014577(n) = Sum_{k=0..n} B(k)*binomial(n,k)  (mod 2)
 
(**)  B(n) = Sum_{k=0..n} A014577(k)*binomial(n,k)  (mod 2)

One way B can be defined using PARI is:

{B(n)=if(k==0,1,if(k==1,0,if(k==2,1,
if((k+1)/2^valuation(k+1,2)==1,1,
if((k+1)/2^valuation(k+1,2)==3,1)))))}
 
Perhaps someone has a concise formula for B that could be used in (*) 
as an alternate formula for A014577(n) ?
 
-- Paul

On Wed, 10 Nov 2004 00:43:09 +0100 Benoit Cloitre <abcloitre at wanadoo.fr>
writes:
> A014577 is an ubiquitous sequence! A little sister of Thue-Morse?
> 
> (i) Kronecker symbol : I already noticed this fact. I had a 
> discussion 
> with Harry Smith on the subject which led him to write something for 
> 
> the Kronecker symbol : http://www.jjj.de/fxt/demo/kronecker.cc at
> 
> A097402(n)= 2*A014577(n-1)-1
> 
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