[seqfan] Fibbinary Numbers and Square Powers of Series: A171791, A263075

Paul D Hanna pauldhanna at juno.com
Fri Oct 9 16:43:31 CEST 2015

Consider the sequences 
both are defined by specific values found for the coefficients of x^(n-1) in the square powers of the g.f., A(x)^(n^2), for n>0. 
Remarkably, both seem to have the same property: 
"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)." 
Sean A. Irvine verified the observation for the initial 1028 terms of A171791,and I have found that it holds for at least the initial 530 terms of A263075. 
Would someone care to verify the observation for A263075 for more terms?  
This makes one wonder what connection exists between fibbinary numbers and square powers of certain power series. 
Is there some deeper relation that is being demonstrated here?   
Thank you, 

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