A006336 - Unexpected Relation to Golden Ratio?

[Paul D. Hanna]

Seqfans, 
     Consider the nice sequence A006336: 
a(n) = a(n-1) + a(n-1 - number of even terms so far). 
http://www.research.att.com/~njas/sequences/A006336
begins:
[1,2,3,5,8,11,16,21,29,40,51,67,88,109,138,167,207,258,309,376,...].
 
My COMMENT (NOT submitted to OEIS): 
-----------------------------------------------------------
It seems that A006336 can be generated by a rule using the golden ratio:
 
a(n) = a(n-1) + a([n/Phi]) for n>1 with a(1)=1  where Phi =
(sqrt(5)+1)/2, 
 
i.e., the number of even terms up to position n-1 equals: 
n-1 - [n/Phi] for n>1 where Phi = (sqrt(5)+1)/2. 
 
(PARI): 
a(n) = if(n==1,1, a(n-1) + a( floor(n/((sqrt(5)+1)/2)) )  )
-----------------------------------------------------------
 
Would someone verify if these are indeed equivalent definitions, at least
empirically?  
Or, what is the first position in which terms are NOT equal? 
 
If these are equivalent, then this is another unexpected appearance of
that ubiquitous constant. 
Thanks, 
     Paul 
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