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<DIV>For sequences like these, the XOR BINOMIAL transform may reveal
some underlying structure.</DIV>
<DIV> </DIV>
<DIV>(For definition of XOR BINOMIAL transform, please see:</DIV>
<DIV><A
href="http://www.research.att.com/projects/OEIS?Anum=A099884">http://www.research.att.com/projects/OEIS?Anum=A099884</A></DIV>
<DIV><A
href="http://www.research.att.com/projects/OEIS?Anum=A099887">http://www.research.att.com/projects/OEIS?Anum=A099887</A></DIV>
<DIV><A
href="http://www.research.att.com/projects/OEIS?Anum=A099888">http://www.research.att.com/projects/OEIS?Anum=A099888</A> )</DIV>
<DIV> </DIV>
<DIV>The XOR BINOMIAL of A014577 is the sequence B:</DIV>
<DIV>B={1,<BR>0,1,<BR>1,0,1,0,<BR>1,0,0,0,1,0,0,0,<BR>1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,<BR>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,<BR>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,...}<BR> </DIV>
<DIV>Perhaps B can be used to find an interesting formula
for A014577,</DIV>
<DIV>using: </DIV>
<DIV> </DIV>
<DIV>(*) A014577(n) = Sum_{k=0..n} B(k)*binomial(n,k) (mod 2)</DIV>
<DIV> </DIV>
<DIV>(**) B(n) = Sum_{k=0..n} A014577(k)*binomial(n,k) (mod 2)</DIV>
<DIV> </DIV>
<DIV>One way B can be defined using PARI is:</DIV>
<DIV> </DIV>
<DIV>{B(n)=if(k==0,1,if(k==1,0,if(k==2,1,<BR>if((k+1)/2^valuation(k+1,2)==1,1,<BR>if((k+1)/2^valuation(k+1,2)==3,1)))))}</DIV>
<DIV> </DIV>
<DIV>Perhaps someone has a concise formula for B that could be used in
(*) </DIV>
<DIV>as an alternate formula for A014577(n) ?</DIV>
<DIV> </DIV>
<DIV>-- Paul</DIV>
<DIV> </DIV>
<DIV>On Wed, 10 Nov 2004 00:43:09 +0100 Benoit Cloitre <<A
href="mailto:abcloitre@wanadoo.fr">abcloitre@wanadoo.fr</A>> writes:<BR>>
A014577 is an ubiquitous sequence! A little sister of Thue-Morse?<BR>>
<BR>> (i) Kronecker symbol : I already noticed this fact. I had a <BR>>
discussion <BR>> with Harry Smith on the subject which led him to write
something for <BR>> <BR>> the Kronecker symbol : <A
href="http://www.jjj.de/fxt/demo/kronecker.cc">http://www.jjj.de/fxt/demo/kronecker.cc</A>
at<BR>> <BR>> A097402(n)= 2*A014577(n-1)-1<BR>> </DIV></BODY></HTML>