[seqfan] A171791: Surprising Conjecture for Odd Terms

[Paul D Hanna]

 SeqFans, 
     Consider the conjecture stated in https://oeis.org/A171791 :  
"It appears that for k>0, 
     a(k) is odd iff k = 2*A003714(n) + 1  for n>=0,  
where A003714 is the fibbinary numbers 
(integers whose binary representation contains no consecutive ones); 
this is true for at least the first 520 terms." 
 
That is, the odd terms are located at positions:  
[0, 1, 3,  5,  9,11,  17,19,21,  33,35,37,41,43,  65,67,69,73,75,81,83,85,  129,131,133,137,139,145,147,149,161,163,165,169,171, ...].  
   
That fibbinary numbers would be involved is surprising since 
the definition of A171791 involves square powers of the g.f.:   
G.f. A(x) satisfies: [x^n] A(x)^((n+1)^2) = 0 for n>1 with a(0)=a(1)=1. 
  
Could someone give a rationale as to why such an unexpected property would hold? 
 
It would also be nice if the conjecture could be tested up to more terms than the initial 520. 
 
Thanks, 
      Paul