CF A110036 vs. A090678 Non-Squashing Partitions mod 2

[Paul D. Hanna]

Seqfans,
        I have verified that A110036 and 2*A090678 coincide, 
at least for the first 4095 terms (n>=1), by  
     |A110036(n)| = 2*A090678(n)   for n>1. 
 
Further, the g.f. for A090678 led me to the following g.f. for A110036:
 
A(x) = (1-x+3*x^2+x^3)/(1+x^2) - 2*Sum_{k>=1}
x^(3*2^(k-1))/Product_{j=0..k} (1+x^(2^j)).
 
The PARI code I used to test this is as follows:
 
{ a(n)=subst(contfrac(1+sum(k=0,#binary(n+1),1/x^(2^k)))[n+1],x,0) }
{
A=(1-x+3*x^2+x^3)/(1+x^2)-2*sum(k=1,12,x^(3*2^(k-1))/prod(j=0,k,1+x^(2^j)
)) }
sum(n=0,4096,a(n)*x^n) - A +O(x^4097)
 
which returns all zeros to verify equivalence. 
 
Paul
 
> Below I paste Neil's "non-squashing" partitions mod 2 (A090678).
>  
> If true, I wonder why.
> Thanks,
>      Paul
> 
> URL:       http://www.research.att.com/projects/OEIS?Anum=A090678
> Sequence:  
> 1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,
>            
> 0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,
>            
> 1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,1,0,
>            1,0,1,0,0,1,1,0,1,0,0,1,0,1,0
> Name:      A088567 mod 2.


> %I A110036
> %S A110036 
> 1,-1,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,-2,0,
> 2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,2,-2,0,2,0,0,
> -2,0,2,-2,0,2,0,-2,0,0,2,0,-2,2,0,0,-2,0,2,0,-2,2,0,-2,0,0,
> 2,0,-2,2,0,0,-2,0,2,-2,0,2,0,-2,0,0,2,-2,0,2,0,0,-2,0,2,0
> %N A110036 Constant term of the partial quotients of the continued 
> fraction 
> expansion of 1 + Sum_{n>=0} 1/x^(2^n), where each partial quotient 
> has the form {x + a(n)} after the initial constant term of 1.
> %C A110036 Suggested by Ralf Stephan.
> %e A110036 1 + 1/x + 1/x^2 + 1/x^4 + 1/x^8 + 1/x^16 + ... =
> [1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].
> %o A110036 (PARI) contfrac(1+sum(n=0,10,1/x^(2^n)))
> %O A110036 0
> %K A110036 ,cofr,sign,
> %A A110036 Paul D Hanna (pauldhanna at juno.com), Jul 08 2005