Product of sigma(k)/phi(k) - CF vs. Non-Squashing Partitions

[Paul D Hanna]

Dear Seqfans,
    Is it merely coincidence that 
 
 |A110036(n)| = 2*A090678(n+3)  ???
 
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.

URL:       http://www.research.att.com/projects/OEIS?Anum=A088567
Sequence:  1,1,1,2,2,3,4,5,6,7,9,10,13,14,18,19,24,25,31,32,40,41,50,
           51,63,64,77,78,95,96,114,115,138,139,163,164,194,195,226,
           227,266,267,307,308,357,358,408,409,471,472,535,536,612,613,
           690,691,785,786,881,882,995,996,1110,1111
Name:      a(n) = number of "non-squashing" partitions of n into distinct parts,
              that is, partitions n=p_1+p_2+...+p_k with 1 <= p_1 < p_2 <
              ... < p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k.




-- "Paul D Hanna" <pauldhanna at juno.com> wrote:

Just now submitted the CF as A110036:
 
%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