A006336 - Unexpected Relation to Golden Ratio?

[Warut Roonguthai]

Very interesting observation, Paul. Your conjecture is too beautiful
to be wrong! However, it seems to be very difficult to prove. I've
checked the first 65,000 terms, but I think it would be easy for some
seqfan programmers to extend the calculation to 10,000,000 terms or so
with C or some other powerful tools. Note that we only have to keep
track of the parity of a(n), not its entire value.

Warut

On 7/22/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
>
> 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