A014577

[Benoit Cloitre]

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

(ii) Shallit's continued fraction :

%S A006466 
1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,1,1,1,1,1,
%N A006466 Continued fraction expansion of C = 2*sum( 1/2^(2^n), n=0 to 
infinity )

consider A081769 the sequence of n such that  A006466(n)=2 :  
5,13,18,23,25,30,38,43,45,53,58,60,65,70,78,83,85,93,98,103,...

then A081769(n) (mod2) = A014577(n-1)  A recent reference explains very 
well this "Dragon" structure for this kind of CF.

(iii) Bernoulli's numbers  :

let a(n)=(2/n)*(1-16^n)*B(4*n)*gcd(2^n,n)

  it is an integer sequence (not in OEIS) :  
1,17,691,929569,221930581,968383680827,2093660879252671,....

then :

a(n) (mod4) = 3-2*A014577(n-1)

Benoit



> It appears that, taking A100004:
> a(n) = (-1/n), where (k/n) is the Kronecker symbol
>
> and replacing +1 with 0 and -1 with 1, we obtain A038189:
> Bit to left of least significant bit in binary expansion of n.
>
> Alternatively, replacing -1 with 0 we obtain (allowing for offset) 
> A014577:
> The regular paper-folding (or dragon curve) sequence.
>
> Jeremy Gardiner
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